Admissible matchings and the Castelnuovo–Mumford regularity of square-free powers

Trung Chau; Kanoy Kumar Das; Amit Roy; Kamalesh Saha

doi:10.1017/nmj.2026.10135

Admissible matchings and the Castelnuovo–Mumford regularity of square-free powers

Part of: Algebraic combinatorics Local rings and semilocal rings Commutative algebra: Homological methods Graph theory

Published online by Cambridge University Press: 03 November 2026

and

Trung Chau*: Affiliation:
Chennai Mathematical Institute , India
Kanoy Kumar Das: Affiliation:
Chennai Mathematical Institute , India School of Basic Sciences, Humanities and Management, Manipal Institute of Technology, MAHE, Manipal, Karnataka, 576104, India
Amit Roy: Affiliation:
Chennai Mathematical Institute , India Department of Mathematics, SRM University-AP, Amaravati 522240, Andhra Pradesh, India
Kamalesh Saha: Affiliation:
Department of Mathematics, SRM University-AP, Amaravati 522240, Andhra Pradesh, India
*: Corresponding author: Trung Chau; Email: chauchitrung1996@gmail.com

Article contents

Abstract
Introduction
Preliminaries
k-admissible matching and a general lower bound
Square-free powers of block graphs
Concluding remarks and some open problems
Funding information
References

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Abstract

Square-free powers of square-free monomial ideals relate homological properties of monomial ideals with combinatorial matching structures. Let $ I $ be any square-free monomial ideal, and let $ \mathcal {H}_I $ denote its associated hypergraph. We refine the concept of $ k $-admissible matching, originally introduced by Erey and Hibi for graphs, by defining a generalized version for arbitrary hypergraphs. This new notion allows us to establish a sharp lower bound for the Castelnuovo–Mumford regularity of the $ k\text {th} $ square-free power $ I^{[k]} $. When $ I $ is equigenerated, the bound is given in terms of a new combinatorial invariant, the $ k $-admissible matching number $ \mathrm {aim}(\mathcal {H}_I, k) $, provided $ I^{[k]} \neq 0 $. Surprisingly, for edge ideals $ I(G) $ of graphs, this gives the first general lower bound on the regularity of $ I(G)^{[k]}$. When $ G $ is a forest, our invariant coincides with the $ k $-admissible matching number of Erey and Hibi. Moreover, we prove that for any block graph $ G $, the bound becomes an exact formula, thereby generalizing the known regularity formula for forests. We conclude by proposing a conjecture on the regularity of square-free powers of edge ideals of chordal graphs.

Keywords

k-admissible matchings square-free powers Castelnuovo–Mumford regularity hypergraph edge ideal

MSC classification

Primary: 05C70: Factorization, matching, partitioning, covering and packing 05E40: Combinatorial aspects of commutative algebra 13D02: Syzygies, resolutions, complexes

Secondary: 05C65: Hypergraphs 13H10: Special types (Cohen-Macaulay, Gorenstein, Buchsbaum, etc.)

Information

Type: Article
Information: Nagoya Mathematical Journal , Volume 261 , 2026 , e57

DOI: https://doi.org/10.1017/nmj.2026.10135 [Opens in a new window]
Creative Commons: This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (https://creativecommons.org/licenses/by/4.0), which permits unrestricted re-use, distribution and reproduction, provided the original article is properly cited.
Copyright: © The Author(s), 2026. Published by Cambridge University Press on behalf of Foundation Nagoya Mathematical Journal

1. Introduction

Square-free monomial ideals form a fundamental class of objects in commutative algebra, with deep connections to combinatorics through their natural association with simplicial complexes and hypergraphs. A central theme of study is to relate the algebraic properties of such ideals to combinatorial invariants of the associated objects. On the other hand, matching theory of graphs and hypergraphs has a rich history and wide range of applications (see, e.g., Bondy and Murty [Reference Bondy and Murty4] and Edmonds [Reference Edmonds10]). For a (simple) hypergraph $\mathcal {H}$ with vertex set $V(\mathcal {H})$ and edge set $E(\mathcal {H})$ , a matching is a collection of pairwise disjoint edges, and the matching number $\nu (\mathcal {H})$ denotes the maximum size of such a collection. Recently, the classical notion of matching has been brought into the realm of combinatorial commutative algebra through the study of square-free powers (see Bigdeli et al. [Reference Bigdeli, Herzog and Zaare-Nahandi3], Crupi et al. [Reference Crupi, Ficarra and Lax6], Das et al. [Reference Das, Roy and Saha8], [Reference Das, Roy and Saha9], Erey et al. [Reference Erey, Herzog, Hibi and Saeedi Madani13], [Reference Erey, Herzog, Hibi and Saeedi Madani14], Ficarra et al. [Reference Ficarra, Herzog and Hibi17], Ficarra and Moradi [Reference Ficarra and Moradi18], Kamberi et al. [Reference Kamberi, Navarra and Qureshi21], Seyed Fakhari [Reference Fakhari23], etc.) and this topic has been of great interest since Erey et al. [Reference Erey, Herzog, Hibi and Saeedi Madani13]. The authors in [Reference Bigdeli, Herzog and Zaare-Nahandi3] introduced the notion of square-free powers for the class of edge ideals of simple graphs. In fact, this notion can be defined for any square-free monomial ideals or edge ideals of hypergraphs. Let $I\subseteq R=\mathbb {K}[x_1,\ldots , x_n]$ be a square-free monomial ideal, and $\mathcal {G}(I)$ be the set of minimal monomial generators of I. Then there is a hypergraph $\mathcal {H}$ such that $I=I(\mathcal {H})$ . The kth square-free power of I, denoted by $I^{[k]}$ , is defined by

$$ \begin{align*} I^{[k]}&=\langle f\in I^k \mid f \text{ is square-free} \rangle\\ &=\langle u_1\dots u_k\mid u_1,\ldots , u_k\in E(\mathcal{H}) \text{ is a matching in } \mathcal{H} \rangle. \end{align*} $$

The study of various algebraic invariants, such as Castelnuovo–Mumford regularity (or simply, regularity), depth, projective dimension, etc., of powers of edge ideals of hypergraphs has been one of the fundamental areas of research in the last few decades. In this context, the notions of matching and induced matching in a hypergraph play a very crucial role. Loosely speaking, a matching in a hypergraph $\mathcal {H}$ corresponds to a regular sequence that is contained in the edge ideal $I(\mathcal {H})$ . On the other hand, the induced matching number of $\mathcal {H}$ , that is, the maximal cardinality of an induced matching, denoted by $\nu _1(\mathcal {H})$ , is an invariant that can be used to estimate lower bounds for the regularity of powers of edge ideals [Reference Banerjee, Chakraborty, Das, Mandal and Selvaraja2]. In particular, the following general lower bound for the regularity of edge ideals of d-uniform hypergraphs is well-known:

$$\begin{align*}\mathrm{reg}(I(\mathcal{H})^k)\geq (d-1)(\nu_1(\mathcal{H})-1)+kd. \end{align*}$$

A relevant question in this direction is the following: Is it possible to find a square-free version of this general lower bound? To the best of our knowledge, there is no general lower bound for the regularity of square-free powers. In Erey et al. [Reference Erey, Herzog, Hibi and Saeedi Madani13], the authors explored this question for the sub-class of edge ideals of graphs. Recall that a graph G is a $2$ -uniform simple hypergraph. The following analogous lower bound for the square-free powers of edge ideals of graphs is provided in [Reference Erey, Herzog, Hibi and Saeedi Madani13, Theorem 2.1]:

$$\begin{align*}\mathrm{reg}(I(G)^{[k]})\geq \nu_1(G)+k \;\; \text{ for all }\; 1\leq k\leq \nu_1(G). \end{align*}$$

In this context, Erey and Hibi [Reference Erey and Hibi15] introduced a new combinatorial notion, called k-admissible matching of a graph G, which can be seen as a generalization of induced matching. The k-admissible matching number of G, denoted by $\mathrm {aim}(G,k)$ , is the maximal size of a k-admissible matching in G. Interestingly, in [Reference Erey and Hibi15, Theorem 25], the authors provided the following upper bound when G is a forest:

$$\begin{align*}\mathrm{reg}(I(G)^{[k]})\leq \mathrm{aim}(G,k)+k \;\; \text{ for all }\; 1\leq k\leq \nu(G). \end{align*}$$

They further showed that this upper bound is attained when $k=2$ and conjectured that this bound will be attained for any $k\leq \nu (G)$ in the case of forests. Later on, Crupi et al. [Reference Crupi, Ficarra and Lax6, Theorem 3.6] settled this conjecture.

In this work, we refine and extend the notion of k-admissible matching of a simple graph G to the realm of hypergraphs (see Definition 3.1). Using this new definition, we provide a general lower bound for square-free powers of any square-free monomial ideal. This extends and strengthens all known lower bounds for square-free powers in the literature.

Theorem 3.11. Let $\mathcal {H}$ be a hypergraph and k an integer where $1\leq k\leq \nu (\mathcal {H})$ . Then for any generalized k-admissible matching M of $\mathcal {H}$ , we have

$$\begin{align*}\mathrm{reg} \left( I(\mathcal{H})^{[k]} \right) \geq |V(M)|-|M|+k. \end{align*}$$

The above-mentioned lower bound is sharp, as seen in Proposition 3.10. In the case of d-uniform hypergraphs, we define a combinatorial invariant, called the k-admissible matching number of $\mathcal {H}$ , denoted by $\mathrm {aim}(\mathcal {H},k)$ (see Definition 3.12), which is analogous to the induced matching number in the context of square-free powers. In particular, $\mathrm {aim}(\mathcal {H},1)$ coincides with $\nu _1(\mathcal {H})$ . As a consequence of our main theorem, we obtain the following lower bound for square-free powers of edge ideals of d-uniform hypergraphs and, in particular, simple graphs.

Theorem 3.16. Let $\mathcal {H}$ be a d-uniform hypergraph and k an integer where $1\leq k\leq \nu (\mathcal {H})$ . Then

$$\begin{align*}\mathrm{reg} \left( I(\mathcal{H})^{[k]} \right) \geq (d-1)\mathrm{aim}(\mathcal{H},k) +k. \end{align*}$$

Specifically, for a simple graph G and for any $1\leq k\leq \nu (G)$ , we have

$$\begin{align*}\mathrm{reg} \left( I(G)^{[k]}\right) \geq \mathrm{aim}(G,k)+k. \end{align*}$$

We provide two classes of simple graphs where we actually get the equality in the formula of the lower bound of the regularity of square-free powers. Indeed, we show that for the class of block graphs, we have the following formula for the regularity.

Theorem 4.11. Let G be a block graph. Then for all $1\leq k\leq \nu (G)$ ,

$$\begin{align*}\mathrm{reg}(I(G)^{[k]})=\mathrm{aim}(G,k)+k. \end{align*}$$

Since the class of block graphs contains the class of forests, the above formula extends the regularity formula of square-free powers of edge ideals of forests provided in Crupi et al. [Reference Crupi, Ficarra and Lax6] and Erey and Hibi [Reference Erey and Hibi15].

Square-free powers are defined as the square-free part of ordinary powers. There is an analog for symbolic powers, first studied by Fakhari [Reference Fakhari24], which we shall call symbolic square-free powers. We remark that an analog of our general lower bound can be obtained for symbolic square-free powers of any square-free monomial ideal [Reference Chau, Das, Roy and Saha5].

This article is organized as follows. In Section 2, we recall the necessary definitions and results from combinatorics and commutative algebra. In Section 3, we prove the general lower bound for square-free powers. In Section 4, we show that the general lower bound is attained for all the square-free powers of edge ideals of block graphs. Finally, Section 5 explores an implication of Theorem 3.11 in the setting of recently introduced matching powers [Reference Erey and Ficarra12] of (not necessarily square-free) monomial ideals. This section also presents a few open problems related to the regularity of square-free powers of edge ideals of various classes of graphs. In particular, building on our results and some computational evidence, we propose the following conjecture.

Conjecture 5.2. Let G be a chordal graph. Then, for all $1\leq k\leq \nu (G)$ , we have

$$\begin{align*}\mathrm{reg}(I(G)^{[k]})=\mathrm{aim}(G,k)+k. \end{align*}$$

2. Preliminaries

In this section, we quickly recall some preliminary notions related to combinatorics and commutative algebra. For any undefined term, the reader is advised to look at Bondy and Murty [Reference Bondy and Murty4], Edmonds [Reference Edmonds10], and Villarreal [Reference Villarreal25].

2.1. Combinatorics

A (simple) hypergraph $\mathcal {H}$ is a pair $\mathcal {H}=(V(\mathcal {H}),E(\mathcal {H}))$ , where $V(\mathcal {H})$ is a finite set, called the vertex set of $\mathcal {H}$ , and $E(\mathcal {H})$ is a collection of subsets of $V(\mathcal {H})$ such that no two elements contain each other, called the edge set of $\mathcal {H}$ . The elements of $V(\mathcal {H})$ are called the vertices of $\mathcal {H}$ , and the elements of $E(\mathcal {H})$ are called the edges of $\mathcal {H}$ . An induced sub-hypergraph of $\mathcal {H}$ on the vertex set $W\subseteq V(\mathcal {H})$ , denoted by $\mathcal {H}[W]$ , is a hypergraph with $V(\mathcal {H}[W])=W$ and $E(\mathcal {H}[W])=\{\mathcal {E}\in E(\mathcal {H})\mid \mathcal {E} \subseteq W\}$ . For $W\subseteq V(\mathcal {H})$ , the hypergraph $\mathcal {H}[V(\mathcal {H})\setminus W]$ is also denoted by $\mathcal {H}\setminus W$ . If $W=\{x\}$ for some $x\in V(\mathcal {H})$ , then $\mathcal {H}\setminus \{x\}$ is simply denoted by $\mathcal {H}\setminus x$ . A matching of $\mathcal {H}$ is a collection of edges M of $\mathcal {H}$ such that $\mathcal {E}\cap \mathcal {E}' =\emptyset $ for any two distinct edges $\mathcal {E},\mathcal {E}'\in M$ . A matching M of $\mathcal {H}$ is called an induced matching of $\mathcal {H}$ if $E(\mathcal {H}[\cup _{\mathcal {E}\in M}\mathcal {E}])=M$ . The induced matching number of $\mathcal {H}$ , denoted by $\nu _1(\mathcal {H})$ , is the maximum cardinality of an induced matching in $\mathcal {H}$ .

A (simple) graph G is nothing but a hypergraph whose edges are of cardinality two. For $A\subseteq V(G)$ , the set of closed neighbors of A is defined as $N_{G}[A]:= A\cup \{x\in V(G)\mid \{x,y\}\in E(G) \text { for some } y\in A\}$ . By a set of open neighbors of A, we mean $N_{G}(A):= N_{G}[A]\setminus A$ . If $A=\{x\}$ , then $N_G(\{x\})$ is simply denoted by $N_G(x)$ . For $x\in V(G)$ , the number $|N_G(x)|$ is called the degree of x and is denoted by $\deg (x)$ . Given two vertices $x,y\in V(G)$ , an induced path of length n between x and y is a collection of vertices $P:a_1,\ldots ,a_n$ such that $a_1=x,a_n=y$ , and $\{a_i,a_j\}\in E(G)$ if and only if $j=i-1$ or $j=i+1$ . A complete graph on n vertices, denoted by $K_n$ , is a graph such that there is an edge between every pair of vertices. A clique of a graph G is a complete induced subgraph of G. A vertex $x\in V(G)$ is said to be a free vertex of G if $G[N_{G}(x)]$ is a complete graph. We say that two edges e and $e'$ of G form a gap in G if $\{e,e'\}$ forms an induced matching in G. The complement of a graph G, denoted by $G^c$ , is a graph such that $V(G^c)=V(G)$ and $E(G^c)=\{\{x,y\}\mid \{x,y\}\not \in E(G)\}$ .

A cycle of length m, denoted by $C_m$ , is a graph such that $V(C_m)=\{x_1,\ldots ,x_m\}$ and $E(C_m)=\{\{x_1,x_m\},\{x_i,x_{i+1}\}\mid 1\le i\le m-1\}$ . A graph G is called a forest if it does not contain any induced cycle. A connected forest is known as a tree. A graph is called chordal if it does not contain an induced $C_n$ for $n\geq 4$ .

2.2. Commutative algebra

For a graded ideal $I\subseteq R$ , the graded minimal free resolution of I is an exact sequence

$$\begin{align*}\mathcal F_{\cdot}: \,\, 0\rightarrow F_p\xrightarrow{\partial_{p}}\cdots\xrightarrow{\partial_{2}} F_1\xrightarrow{\partial_1} F_0\xrightarrow{\partial_0} I\rightarrow 0, \end{align*}$$

where $F_i=\oplus _{j\in \mathbb N}R(-j)^{\beta _{i,j}(I)}$ is a free R-module for each $i\ge 0$ and $R(-j)$ denotes the polynomial ring R with grading shifted by j. The numbers $\beta _{i,j}(I)$ are called the ith $\mathbb N$ -graded Betti numbers of I in degree j. The Castelnuovo–Mumford regularity (in short, regularity) of I, denoted by $\mathrm {reg}(I)$ , is defined as $\mathrm {reg}(I):= \max \{j-i\mid \beta _{i,j}(I)\neq 0\}$ . If I is the zero ideal, then we adopt the convention that $\mathrm {reg}(I)=0$ . The following two well-known results regarding the regularity of a graded ideal will be used frequently in the subsequent sections.

Lemma 2.1 ([Reference Dao, Huneke and Schweig7, Lemma 2.10]).

Let $I \subseteq R$ be a monomial ideal, m be a monomial of degree d in R, and x be an indeterminate in R. Then

(i) $\mathrm {reg}(I+\langle x\rangle )\le \mathrm {reg}(I) $ ;
(ii) $\mathrm {reg}(I:x)\le \mathrm {reg}(I)$ ;
(iii) $\mathrm {reg} I \le \max \{\mathrm {reg}(I : m) + d, \mathrm {reg}( I+ \langle m\rangle )\}$ .

Lemma 2.2 (Villarreal [Reference Villarreal25]).

Let $I_1\subseteq R_1=\mathbb K[x_1,\ldots ,x_m]$ and $I_2\subseteq R_2=\mathbb K[x_{m+1},\ldots , x_n]$ be two graded ideals. Consider the ideal $I=I_1R+I_2R\subseteq R=\mathbb K[x_1,\ldots ,x_n]$ . Then

$$\begin{align*}\mathrm{reg}(I)=\mathrm{reg}(I_1)+\mathrm{reg}(I_2)-1. \end{align*}$$

Let $\mathcal {H}$ be a hypergraph on the vertex set $V(\mathcal {H})=\{x_1,\ldots ,x_n\}$ . Identifying the vertices of $\mathcal {H}$ as indeterminates, we consider the polynomial ring $R=K[x_1,\ldots ,x_n]$ over a field K. Corresponding to any $F\subseteq V(\mathcal {H})$ , we denote the square-free monomial $\prod _{x_i\in F}x_i$ of R by $\mathbf x_{F}$ . The edge ideal of $\mathcal {H}$ , denoted by $I(\mathcal {H})$ , is a square-free monomial ideal of R defined as

$$\begin{align*}I(\mathcal{H}):= \langle \mathbf x_{\mathcal{E}}\mid \mathcal{E}\in E(\mathcal{H})\rangle. \end{align*}$$

Note that any square-free monomial ideal can be seen as the edge ideal of a hypergraph.

3. k-admissible matching and a general lower bound

The induced matching number of a hypergraph $\mathcal {H}$ is a combinatorial invariant that has been used to estimate lower bounds for the regularity of ordinary and symbolic powers of the edge ideal $I(\mathcal {H})$ [Reference Banerjee, Chakraborty, Das, Mandal and Selvaraja2]. In the case of square-free powers, a similar attempt has been made in [Reference Erey, Herzog, Hibi and Saeedi Madani13, Theorem 2.1], where the authors have shown that $\mathrm {reg}(I(G)^{[k]})\geq \nu _1(G)+k$ for all $1\leq k\leq \nu _1(G)$ . Later on, as a generalization of induced matching, k-admissible matching was introduced by Erey and Hibi. Then it was proved that for the class of forests, the equality $\mathrm {reg}(I(G)^{[k]})=\mathrm {aim}(G,k)+k$ holds for all $1\leq k\leq \nu (G)$ (see [Reference Crupi, Ficarra and Lax6, Theorem 3.6] and [Reference Erey and Hibi15, Theorem 25]). Furthermore, it follows from the work of Fakhari [Reference Fakhari23, Theorem 4.3], that for a Cameron–Walker graph G, $\mathrm {reg}(I(G)^{[k]})=\mathrm {aim}(G,k)+k$ for all $1\leq k\leq \nu (G)$ . Based on these results, one can expect that for an arbitrary graph G, the regularity of $I(G)^{[k]}$ is bounded below by the combinatorial invariant $\mathrm {aim}(G,k)+k$ .

Inspired by this idea, in this section, we refine and extend their definition of k-admissible matching for simple graphs to all hypergraphs. To avoid any ambiguity, we call this refined notion of matching to be generalized k-admissible matching of a hypergraph and use the same notations. We use $\sqcup $ to denote disjoint unions. For a matching M, let $V(M)$ denote the set of vertices that belong to the hyperedges in M.

Definition 3.1. Let $\mathcal {H}$ be a hypergraph and M a matching of $\mathcal {H}$ . Then for any $1\leq k\leq \nu (\mathcal {H})$ , we say that M is a generalized k-admissible matching if there is a partition $M=M_1\sqcup \cdots \sqcup M_r$ such that

1. for any edge $\mathcal {E}\in \mathcal {H}[V(M)]$ , we have $\mathcal {E}\in E(\mathcal {H}[V(M_i)])$ for some $i\in [r]$ ;
2. $k\leq |M|=\sum _{i=1}^r |M_i|\leq r+k-1$ ;
3. for any $i\in [r]$ and any matching $M_i'$ in $\mathcal {H}[V(M_i)]$ with $|M_i'|=|M_i|$ , one has $V(M_i')=V(M_i)$ .

3.1. Existence of generalized k-admissible matching

Let $\mathcal {H}$ be a hypergraph and $1\leq k\leq \nu (\mathcal {H})$ . Take a matching $M'$ of $\mathcal {H}$ with $\vert M'\vert =k$ . Now, consider the induced sub-hypergraph $\mathcal {H}'=\mathcal {H}[V(M')]$ and pick a minimal generator u of the ideal $I(\mathcal {H}')^{[k]}$ . Then the induced sub-hypergraph $\mathcal {H}[\mathrm {supp}(u)]$ has a matching number k, and any matching of size k in $\mathcal {H}[\mathrm {supp}(u)]$ consists of all the vertices in $\mathrm {supp}(u)$ . Thus, any matching M with $|M|=k$ in $\mathcal {H}[\mathrm {supp}(u)]$ will give a generalized k-admissible matching of $\mathcal {H}$ by taking the partition $M=M_1$ .

Our aim in this section is to give a lower bound for $\mathrm {reg} I(\mathcal {H})^{[k]}$ for any $1\le k\le \nu (\mathcal {H})$ using generalized k-admissible matching. We present some basic observations for this new concept.

Remark 3.2. Let $\mathcal {H}$ be a hypergraph and $M\subseteq E(\mathcal {H})$ be a matching of $\mathcal {H}$ .

1. A matching M of $\mathcal {H}$ is a generalized $1$ -admissible matching if and only if M is an induced matching of $\mathcal {H}$ .
2. Condition $(3)$ in Definition 3.1 implies that $\nu (\mathcal {H}[V(M_i)])=|M_i|$ for any $i\in [r]$ . Indeed, suppose otherwise that there exists a matching $M'$ in $\mathcal {H}[V(M_i)]$ such that $|M'|>|M_i|$ . Then consider a proper subset $M"$ of $M'$ such that $|M"|=|M_i|$ . This contradicts Definition 3.1(3), as desired.
3. Condition $(3)$ in Definition 3.1 is equivalent to the following fact: for any $i\in [r]$ , $I(\mathcal {H}[V(M_i)])^{[|M_i|]}$ is generated by one monomial $\prod _{x\in V(M_i)} x$ . This is immediate from the previous remark and Definition 3.1(3).

The following result follows from verifying the conditions in Definition 3.1; thus, we state it without proof.

Proposition 3.3. Let $\mathcal {H}$ be a hypergraph and $\mathcal {H}_1$ an induced sub-hypergraph of $\mathcal {H}$ . Then, any generalized k-admissible matching in $\mathcal {H}_1$ is a generalized k-admissible matching of $\mathcal {H}$ .

Our primary tool to establish the regularity lower bound is the idea of Betti splitting introduced in Francisco et al. [Reference Francisco, Ha and Van Tuyl19].

Definition 3.4 ([Reference Francisco, Ha and Van Tuyl19, Definition 1.1]).

Let $I,J, \text { and } K$ be monomial ideals such that $\mathcal {G}(I)$ is the disjoint union of $\mathcal {G}(J)\text { and } \mathcal {G}(K)$ . Then $I=J+K$ is a Betti splitting if

(1) $$ \begin{align} \beta_{i,j}(I)=\beta_{i,j}(J)+\beta_{i,j}(K)+\beta_{i-1,j}(J\cap K) \;\;\;\text{ for all } i,j. \end{align} $$

Although the notion of Betti splitting was formally introduced in Francisco et al. [Reference Francisco, Ha and Van Tuyl19], it first appeared in the work of Eliahou and Kervaire [Reference Eliahou and Kervaire11]. In fact, in this article, we use the concept of a splittable monomial ideal as defined in Eliahou and Kervaire [Reference Eliahou and Kervaire11].

Definition 3.5 (cf. Eliahou and Kervaire [Reference Eliahou and Kervaire11]).

A monomial ideal I is said to be splittable if $I=J+K$ , where J and K are two non-zero ideals and $\mathcal {G} (I)=\mathcal {G}(J)\sqcup \mathcal {G}(K)$ , such that there exists a splitting function

$$ \begin{align*} (\phi,\psi): \mathcal{G}(J\cap K)&\rightarrow \mathcal{G}(J)\times \mathcal{G}(K)\\ w &\mapsto (\phi(w),\psi(w)), \end{align*} $$

which satisfies the following two properties:

(a) for each $w\in \mathcal {G}(J\cap K)$ , $w=\mathrm {lcm} (\phi (w),\psi (w))$ ;
(b) for each $S\subseteq \mathcal {G}(J\cap K)$ , both $\mathrm {lcm}(\phi (S))$ and $\mathrm {lcm}(\psi (S))$ strictly divide $\mathrm {lcm}(S)$ .

Remark 3.6. The fact that for a splittable monomial ideal $I=J+K$ in Definition 3.5, Equation (1) holds, follows from [Reference Fatabbi16, Proposition 3.2].

It is evident from Equation (1) that Betti splittings allow us to study the graded Betti numbers of a splittable monomial ideal I in terms of the graded Betti numbers of the smaller ideals J and K. We provide an important application of this below.

Lemma 3.7. Let $\{I_i\}_{i\in \mathbb {N}}$ be a sequence of proper square-free monomial ideals in $\mathbb {K}[x_1,\dots , x_{N_1}]$ such that for any $i\in \mathbb {N}$ and any $f\in I_i$ , we have $f/x\in I_{i-1}$ for any $x\in \mathrm {supp}(f)$ . Let $\{L_i\}_{i\in \mathbb {N}}$ be a sequence of square-free monomial ideals in $\mathbb {K}[y_1,\dots , y_{N_2}]$ with the same condition. Let $n,k,l$ be three integers where $l<k\leq n$ . Set

$$ \begin{align*} J&:= I_k L_{n-k},\\ K&:= I_{k-1} L_{n-k+1} + \cdots + I_lL_{n-l}. \end{align*} $$ StartLayout 1st Row 1st Column upper J 2nd Column colon equals upper I Subscript k Baseline upper L Subscript n minus k Baseline comma 2nd Row 1st Column upper K 2nd Column colon equals upper I Subscript k minus 1 Baseline upper L Subscript n minus k plus 1 Baseline plus midline horizontal ellipsis plus upper I Subscript l Baseline upper L Subscript n minus l Baseline period EndLayout

Assume that $J\neq 0$ . Then $J+K$ is a Betti splitting. In particular, we have

$$\begin{align*}\mathrm{reg}\left( J+K \right) \geq \max \{ \mathrm{reg} \left( I_k \right) + \mathrm{reg}\left( L_{n-k} \right),\ \mathrm{reg} \left( I_k\right) + \mathrm{reg} \left( L_{n-k+1} \right) -1 \}. \end{align*}$$

Proof. First, we show that $\mathcal {G}(J)\cap \mathcal {G}(K)=\emptyset $ . For this, it suffices to show that whenever $I_i$ is non-zero for some $i\in \mathbb N$ , then $I_i\subsetneq I_{i-1}$ , as an analogous statement for $L_i\neq \langle 0\rangle $ would follow. Indeed, let $f\in \mathcal {G}(I_i)$ , then $f/x\in I_{i-1}$ by our hypotheses for any $x\in \mathrm {supp}(f)$ . Since $f\in \mathcal {G}(I_i)$ , we have $f/x\notin I_{i}$ . Thus, we have $\mathcal {G}(J)\cap \mathcal {G}(K)=\emptyset $ .

Note that if $K=\langle 0\rangle $ , then $J=J+K$ is trivially a Betti splitting, and we have the result. Thus, we may assume $K\neq \langle 0\rangle $ . We next show that $J\cap K= I_k L_{n-k+1}$ . Indeed, the reverse inclusion is clear. For the inclusion, since $I_i$ and $L_j$ share no variable for any i and j, any minimal monomial generator of $J\cap K$ is of the form $\mathrm {lcm}(fg,f'g'),$ where $f\in I_k, g\in L_{n-k}, f'\in I_{r},$ and $g'\in L_{n-r}$ for some $l \leq r<k$ . It is then clear that $\mathrm {lcm}(fg,f'g')\in I_kL_{n-k+1}$ , and thus the inclusion follows.

Let $(>)$ be a fixed monomial ordering on $\{x_1,\dots , x_{N_1}, y_1,\dots y_{N_2}\}$ and $(>_{lex})$ be the lex ordering on the monomials induced from $(>)$ .

Consider any generator $f\in \mathcal {G}(J\cap K) = \mathcal {G}(I_kL_{n-k+1})$ . Set $f=f_1f_2$ , where $f_1\in \mathcal {G}(I_k)$ and $f_2\in \mathcal {G}(L_{n-k+1})$ . Set

$$ \begin{align*} x_f&:= \max_{>} \mathrm{supp}(f_1),\\ y_f&:= \max_{>} \mathrm{supp}(f_2),\\ f_I &:= \max_{>_{lex}} \{ g \colon x_f \mid g \text{ and } g\mid f_1, \text{ and } f_1/g\in \mathcal{G}(I_{k-1}) \},\\ f_{L} &:= \max_{>_{lex}} \{g \colon y_f \mid g \text{ and } g\mid f_2, \text{ and } f_2/g\in \mathcal{G}(L_{n-k} ) \}. \end{align*} $$

StartLayout 1st Row 1st Column x Subscript f 2nd Column colon equals ModifyingBelow max With greater than supp left parenthesis f 1 right parenthesis comma 2nd Row 1st Column y Subscript f 2nd Column colon equals ModifyingBelow max With greater than supp left parenthesis f 2 right parenthesis comma 3rd Row 1st Column f Subscript upper I 2nd Column colon equals max Underscript greater than Subscript l e x Baseline Endscripts left brace g colon x Subscript f Baseline StartAbsoluteValue g and g EndAbsoluteValue f 1 comma and f 1 divided by g element of script upper G left parenthesis upper I Subscript k minus 1 Baseline right parenthesis right brace comma 4th Row 1st Column f Subscript upper L 2nd Column colon equals max Underscript greater than Subscript l e x Baseline Endscripts left brace g colon y Subscript f Baseline StartAbsoluteValue g and g EndAbsoluteValue f 2 comma and f 2 divided by g element of script upper G left parenthesis upper L Subscript n minus k Baseline right parenthesis right brace period EndLayout

The existence of $f_I$ and $f_L$ is due to the hypotheses.

Let $(\phi ,\psi )$ define a map as follows:

$$ \begin{align*} (\phi,\psi) \colon \mathcal{G}(J\cap K) &\to \mathcal{G}(J) \times \mathcal{G}(K)\\ f&\mapsto \left( \frac{f}{f_{L}} , \frac{f}{f_I} \right). \end{align*} $$

StartLayout 1st Row 1st Column left parenthesis phi comma psi right parenthesis colon script upper G left parenthesis upper J intersection upper K right parenthesis 2nd Column right arrow script upper G left parenthesis upper J right parenthesis times script upper G left parenthesis upper K right parenthesis 2nd Row 1st Column f 2nd Column right arrow from bar left parenthesis StartFraction f Over f Subscript upper L Baseline EndFraction comma StartFraction f Over f Subscript upper I Baseline EndFraction right parenthesis period EndLayout

Due to our constructions, the map is well-defined. We show that $(\phi ,\psi )$ indeed defines a splitting function:

(a) for each $f\in \mathcal {G}(J\cap K)$ , we have $f=\mathrm {lcm}(\frac {f}{f_L},\frac {f}{f_{I}}) = \mathrm {lcm}(\phi (f),\psi (f))$ since $f_L$ and $f_{I}$ have disjoint support;
(b) for each $S =\{h_1,\dots , h_s\} \subseteq \mathcal {G}(J\cap K)$ , both $\mathrm {lcm}(\phi (S)) $ and $\mathrm {lcm}(\psi (S))$ strictly divide $\mathrm {lcm}(S)$ since by design, we have
(1) $$ \begin{align} x_S,y_S\mid \mathrm{lcm}(h_1,\dots, h_s) &= \mathrm{lcm}(S), \end{align} $$

(2) $$ \begin{align} x_S\nmid \mathrm{lcm}\left(\frac{h_1}{{h_1}_I}, \dots, \frac{h_s}{{h_s}_I}\right) &= \mathrm{lcm}(\phi(S)), \end{align} $$

(3) $$ \begin{align} y_S\nmid \mathrm{lcm}\left(\frac{h_1}{{h_1}_{L}}, \dots, \frac{h_s}{{h_s}_{L}}\right) &= \mathrm{lcm}(\psi(S)), \end{align} $$

where
$$ \begin{align*} x_S:= \max_{>} \{ x_{{h_i}} \colon i\in [s] \},\\ y_S:= \max_{>} \{ y_{{h_i}} \colon i\in [s] \}. \end{align*} $$

Indeed, (1) follows from the definition of $x_S$ and $y_S$ . Regarding (2), observe that if $x_S\in \mathrm {supp}(h_i)$ for some $i\in [s]$ , then we have $x_S=x_{h_i}\mid {h_i}_I$ . This implies that $x_S\nmid \frac {h_i}{{h_i}_I}$ for any $i\in [s]$ . Thus, (2) follows. The proof of (3) is similar by symmetry.

Thus, $I=J+K$ is a Betti splitting. The last statement follows from Equation (1).

We now apply Lemma 3.7 in the special case of square-free powers of edge ideals to obtain an important regularity formula needed for the general lower bound of square-free powers. Here, we adopt the convention that $I^{[l]}=R$ for any $l\le 0$ .

Lemma 3.8. Let $\mathcal {H}$ and $\mathcal {H}'$ be hypergraphs that have at least one edge each and no common vertex, and k a positive integer such that $\nu (\mathcal {H})+1\leq k\leq \nu (\mathcal {H})+ \nu (\mathcal {H}')$ . Then

$$\begin{align*}I(\mathcal{H}\sqcup \mathcal{H}')^{[k]} = J + K, \end{align*}$$

where

$$ \begin{align*} J&= I(\mathcal{H})^{[\nu(\mathcal{H})]} I(\mathcal{H}')^{[k-\nu(\mathcal{H})]} , \\ K&= I(\mathcal{H})^{[\nu(\mathcal{H})-1]} I(\mathcal{H}')^{[k-\nu(\mathcal{H})+1]} + \cdots + I(\mathcal{H})^{[k-\nu(\mathcal{H}')]} I(\mathcal{H}')^{[\nu(\mathcal{H}')]}, \end{align*} $$

is a Betti splitting. In particular, we have

$$ \begin{align*} \mathrm{reg}\left( I(\mathcal{H}\sqcup \mathcal{H}')^{[k]} \right) \geq \max \{ \mathrm{reg} \left( I(\mathcal{H})^{[\nu(\mathcal{H})]} \right) + & \ \mathrm{reg}\left( I(\mathcal{H}')^{[k-\nu(\mathcal{H})]}\right),\\ & \mathrm{reg} \left( I(\mathcal{H})^{[\nu(\mathcal{H})]} \right) + \mathrm{reg} \left( I(\mathcal{H}')^{[k-\nu(\mathcal{H})+1]} \right) -1 \}. \end{align*} $$

StartLayout 1st Row 1st Column reg left parenthesis upper I left parenthesis script upper H square cup script upper H Superscript prime Baseline right parenthesis Superscript left bracket k right bracket Baseline right parenthesis greater than or equals max left brace reg left parenthesis upper I left parenthesis script upper H right parenthesis Superscript left bracket nu left parenthesis script upper H right parenthesis right bracket Baseline right parenthesis plus 2nd Column reg left parenthesis upper I left parenthesis script upper H Superscript prime Baseline right parenthesis Superscript left bracket k minus nu left parenthesis script upper H right parenthesis right bracket Baseline right parenthesis comma 2nd Row 1st Column Blank 2nd Column reg left parenthesis upper I left parenthesis script upper H right parenthesis Superscript left bracket nu left parenthesis script upper H right parenthesis right bracket Baseline right parenthesis plus reg left parenthesis upper I left parenthesis script upper H Superscript prime Baseline right parenthesis Superscript left bracket k minus nu left parenthesis script upper H right parenthesis plus 1 right bracket Baseline right parenthesis minus 1 right brace period EndLayout

Proof. The condition $\nu (\mathcal {H})+1\leq k\leq \nu (\mathcal {H})+ \nu (\mathcal {H}')$ implies that $J\neq \langle 0\rangle $ . The claim that $I(\mathcal {H}\sqcup \mathcal {H}')^{[k]}=J+K$ follows immediately from definition. It is clear that for any monomial $f\in I(\mathcal {H})^{[k]}$ , we have $f/x\in I(\mathcal {H})^{[k-1]}$ for any $x\in \mathrm {supp}(f)$ . The results then follow from Lemma 3.7.

We are now ready to prove the main result of this section. Before that, we recall the result [Reference Erey, Herzog, Hibi and Saeedi Madani13, Corollary 1.3] on the regularity in the context of taking induced subgraphs. Note that while the authors stated the result for only simple graphs, their proof also works for any hypergraph. Thus, we have the following.

Lemma 3.9. Let $\mathcal {H}$ be a hypergraph, $\mathcal {H}_1$ an induced sub-hypergraph of $\mathcal {H}$ , and k a positive integer. Then

$$ \begin{align*} \mathrm{reg}\left( I(\mathcal{H}_1)^{[k]} \right) \leq \mathrm{reg}\left( I(\mathcal{H})^{[k]} \right). \end{align*} $$

We first consider the special case when hyperedges of $\mathcal {H}$ are pairwise disjoint.

Proposition 3.10. Let $\mathcal {H}$ be a hypergraph such that any two hyperedges of $\mathcal {H}$ have an empty intersection. Then for any integer k where $1\leq k\leq \nu (\mathcal {H})$ , we have

$$ \begin{align*} \mathrm{reg} \left( I(\mathcal{H})^{[k]} \right) &= \max\{ |V(M)|-|M|\colon M \text{ is a generalized } k\text{-admissible matching of } \mathcal{H}\}+k\\ &= |V(\mathcal{H})| - |E(\mathcal{H})|+k. \end{align*} $$

StartLayout 1st Row 1st Column reg left parenthesis upper I left parenthesis script upper H right parenthesis Superscript left bracket k right bracket Baseline right parenthesis 2nd Column equals max left brace StartAbsoluteValue upper V left parenthesis upper M right parenthesis EndAbsoluteValue minus StartAbsoluteValue upper M EndAbsoluteValue colon upper M is a generalized k minus admissible matching of script upper H right brace plus k 2nd Row 1st Column Blank 2nd Column equals StartAbsoluteValue upper V left parenthesis script upper H right parenthesis EndAbsoluteValue minus StartAbsoluteValue upper E left parenthesis script upper H right parenthesis EndAbsoluteValue plus k period EndLayout

Proof. Set $E(\mathcal {H})=\{\mathcal {E}_1,\dots , \mathcal {E}_q\}$ , and for any $i\in [q]$ , set $m_i=\prod _{x\in \mathcal {E}_i} x$ . As any two hyperedges of $\mathcal {H}$ have empty intersection, the monomials $\{m_i\}_{i=1,\dots , q}$ have pairwise disjoint supports. In particular, $I(\mathcal {H})=(m_1,\dots , m_q)$ is a complete intersection. It is straightforward that $E(\mathcal {H})$ itself is a generalized k-admissible matching for any $1\leq k\leq \nu (\mathcal {H})$ . Therefore, we have

$$\begin{align*}\max\{ |V(M)|-|M|\colon M \text{ is a } k\text{-admissible matching of } \mathcal{H}\} = |V(E(\mathcal{H}))| - q = \sum_{i=1}^q |\mathcal{E}_i| - q. \end{align*}$$

It now suffices to show that

$$\begin{align*}\mathrm{reg} \left( I(\mathcal{H})^{[k]} \right) = \sum_{i=1}^q |\mathcal{E}_i| - q+k, \end{align*}$$ reg left parenthesis upper I left parenthesis script upper H right parenthesis Superscript left bracket k right bracket Baseline right parenthesis equals sigma summation Underscript i equals 1 Overscript q Endscripts StartAbsoluteValue script upper E Subscript i Baseline EndAbsoluteValue minus q plus k comma

by induction on q and k. If $q=1$ , then we must have $k=1$ . The result then follows immediately. On the other hand, if $k=1$ , then as $I(\mathcal {H})^{[k]}$ is a complete intersection, it is known that

$$\begin{align*}\mathrm{reg} \left( I(\mathcal{H})^{[k]} \right) = \mathrm{reg} \left( I(\mathcal{H}) \right) = \sum_{i=1}^q \deg m_i - (q-1) = \sum_{i=1}^q |\mathcal{E}_i| - q+k, \end{align*}$$

as desired. We can now assume that $q,k\geq 2$ . Set $J:= m_q(m_1,\dots , m_{q-1})^{[k-1]}$ and $K:= (m_1,\dots , m_{q-1})^{[k]}$ . It is straightforward that $J\cap K = m_q (m_1,\dots , m_{q-1})^{[k]}$ . By Lemma 3.8, $I(\mathcal {H})^{[k]}=J+K$ is a Betti splitting, and thus by Definition 3.4, we have

$$ \begin{align*} \mathrm{reg} \left( I(\mathcal{H})^{[k]} \right) = \max\{ \mathrm{reg} & \left( m_q(m_1,\dots, m_{q-1})^{[k-1]} \right) , \quad \mathrm{reg} \left( (m_1,\dots, m_{q-1})^{[k]} \right) ,\\ & \mathrm{reg} \left( m_q(m_1,\dots, m_{q-1})^{[k]} \right) -1 \}. \end{align*} $$

By the induction hypotheses, we have

$$ \begin{align*} \mathrm{reg} \left( (m_1,\dots, m_{q-1})^{[k-1]} \right) &= \sum_{i=1}^{q-1} |\mathcal{E}_i| - (q-1)+(k-1) = \sum_{i=1}^{q-1} |\mathcal{E}_i| - q+k,\\ \mathrm{reg} \left( (m_1,\dots, m_{q-1})^{[k]} \right) &= \sum_{i=1}^{q-1} |\mathcal{E}_i| - (q-1)+k = \sum_{i=1}^{q-1} |\mathcal{E}_i| - q+k+1. \end{align*} $$

StartLayout 1st Row 1st Column reg left parenthesis left parenthesis m 1 comma ellipsis comma m Subscript q minus 1 Baseline right parenthesis Superscript left bracket k minus 1 right bracket Baseline right parenthesis 2nd Column equals sigma summation Underscript i equals 1 Overscript q minus 1 Endscripts StartAbsoluteValue script upper E Subscript i Baseline EndAbsoluteValue minus left parenthesis q minus 1 right parenthesis plus left parenthesis k minus 1 right parenthesis equals sigma summation Underscript i equals 1 Overscript q minus 1 Endscripts StartAbsoluteValue script upper E Subscript i Baseline EndAbsoluteValue minus q plus k comma 2nd Row 1st Column reg left parenthesis left parenthesis m 1 comma ellipsis comma m Subscript q minus 1 Baseline right parenthesis Superscript left bracket k right bracket Baseline right parenthesis 2nd Column equals sigma summation Underscript i equals 1 Overscript q minus 1 Endscripts StartAbsoluteValue script upper E Subscript i Baseline EndAbsoluteValue minus left parenthesis q minus 1 right parenthesis plus k equals sigma summation Underscript i equals 1 Overscript q minus 1 Endscripts StartAbsoluteValue script upper E Subscript i Baseline EndAbsoluteValue minus q plus k plus 1 period EndLayout

Therefore, we have

$$ \begin{align*} \mathrm{reg} \left( I(\mathcal{H})^{[k]} \right) &= \max\left\{\mathrm{reg}\left( m_q\right)+ \mathrm{reg} \left((m_1,\dots, m_{q-1})^{[k-1]} \right) ,\quad \mathrm{reg} \left( (m_1,\dots, m_{q-1})^{[k]} \right) ,\right.\\ &\left.\quad\mathrm{reg}\left( m_q\right) +\mathrm{reg} \left((m_1,\dots, m_{q-1})^{[k]} \right) -1 \right\} \\ &=\max\left\{ \sum_{i=1}^{q} |\mathcal{E}_i| - q+k ,\quad \sum_{i=1}^{q-1} |\mathcal{E}_i| - q+k+1,\quad \sum_{i=1}^{q} |\mathcal{E}_i| - q+k \right\}\\ &= \sum_{i=1}^{q} |\mathcal{E}_i| - q+k, \end{align*} $$

as desired. This concludes the proof.

We are now ready to prove our main theorem of this section.

Theorem 3.11. Let $\mathcal {H}$ be a hypergraph and k an integer where $1\leq k\leq \nu (\mathcal {H})$ . Then for any generalized k-admissible matching M of $\mathcal {H}$ , we have

$$\begin{align*}\mathrm{reg} \left( I(\mathcal{H})^{[k]} \right) \geq |V(M)|-|M|+k. \end{align*}$$

Consequently, we have

$$\begin{align*}\mathrm{reg} \left( I(\mathcal{H})^{[k]} \right) \geq \max\{ |V(M)|-|M|\colon M \text{ is a generalized } k\text{-admissible matching of } \mathcal{H}\}+k. \end{align*}$$

Proof. Without loss of generality, assume that $\mathcal {H}$ has no isolated vertex. We induct on k and $|E(\mathcal {H})|$ . If $k=1$ , then the result follows from [Reference Morey and Villarreal22, Corollary 3.9]. If $\mathcal {H}$ has one edge, then we must have $k=1$ , and the result follows trivially. By induction, we can assume that $k\geq 2$ and $\mathcal {H}$ has at least two edges, and whenever $k'<k$ or $|E(\mathcal {H}')|<|E(\mathcal {H})|$ , we have

$$\begin{align*}\mathrm{reg} \left( I(\mathcal{H}')^{[k']} \right) \geq |V(M')|-|M'|+k', \end{align*}$$

where $M'$ is any generalized $k'$ -admissible matching of $\mathcal {H}'$ . Now assume that $M=\sqcup _{i=1}^r M_i$ is a generalized k-admissible matching of $\mathcal {H}$ where the conditions in Definition 3.1 hold. Set $a=|V(M)| - |M|$ , and $a_i=|M_i|$ for any $i\in [r]$ . By Lemma 3.9, it suffices to show that

(2) $$ \begin{align} \mathrm{reg}\left( I(\mathcal{H}[V(M)])^{[k]} \right) \geq a+k. \end{align} $$

Suppose that $r=1$ . Then $k\leq |M|=a_1\leq k$ , and thus $|M|=a_1=k$ . Since the ideal $I(\mathcal {H}[V(M)])^{[k]}$ is the principal ideal generated by the product of all variables in M, we have

$$\begin{align*}\mathrm{reg}\left( I(\mathcal{H}[V(M)])^{[k]} \right) = |V(M)|= \left(|V(M)| - |M|\right)+ |M| = a+k, \end{align*}$$

as desired. Now we can assume that $r\geq 2$ . Due to condition (1) of Definition 3.1, the graph $\mathcal {H}[V(M)]$ is exactly the disjoint union of the hypergraphs $\mathcal {H}[V(M_i)]$ where i ranges in $[r]$ . Set $\mathcal {H}_1:= \mathcal {H}[V(M_1)]$ , $M'=\sqcup _{i=2}^r M_i$ , and $\mathcal {H}_2:= \mathcal {H}[V(M')] = \sqcup _{i=2}^r \mathcal {H}[V(M_i)]$ . Then $\mathcal {H}_1$ and $\mathcal {H}_2$ share no vertex. If $a_i=1$ for any $i\in [r]$ , then M is a matching whose hyperedges are pairwise disjoint. The result then follows from Proposition 3.10. We can now assume that $a_r\geq 2$ . We then have $k-\nu (H_1)=k-a_1\geq \sum _{i=2}^r a_i - (r-1)> 0$ and $k\leq a_1+ \sum _{i=2}^ra_i=\nu (H_1)+\nu (H_2)$ by conditions (2) and (3) of Definition 3.1. Thus, by Lemma 3.8, we have

(3) $$ \begin{align} \mathrm{reg} \left( I(\mathcal{H}[V(M)]) \right)^{[k]} \geq \max \{ |V(M_1)| + \mathrm{reg} \left( I(\mathcal{H}_2)^{[k-a_1]} \right), \quad |V(M_1)| + \mathrm{reg} \left( I(\mathcal{H}_2)^{[k-a_1+1]} \right) -1 \}, \end{align} $$

where we used the fact that $\mathrm {reg} \left ( I(\mathcal {H}_1)^{[a_1]}\right )$ is the principal ideal generated by a monomial of degree $|V(M_1)|$ from condition (3) of Definition 3.1. Recall that we have $|M|=\sum _{i=1}^r a_i \leq r+k-1$ . If $|M|<r+k-1$ , then

$$\begin{align*}|M'| = \sum_{i=2}^r a_i = |M|-a_1 < (r+k-1)-a_1 = (r-1)+ (k-a_1), \end{align*}$$

and hence

$$\begin{align*}|M'| \leq (r-1) + (k-a_1) - 1. \end{align*}$$

On the other hand, since $k\leq |M|=a_1+|M'|$ , we have $k-a_1\leq |M'|$ . Thus, $M'=\sqcup _{i=2}^r M_i$ is a generalized $(k-a_1)$ -admissible matching in $\mathcal {H}_2$ . By induction, we have

$$ \begin{align*} \mathrm{reg}\left( I(\mathcal{H}_2)^{[k-a_1]} \right) \geq (|V(M')| - |M'|) + (k-a_1) &= (a-|V(M_1)| + a_1) + (k-a_1) \\ &= a -|V(M_1)|+ k. \end{align*} $$

StartLayout 1st Row 1st Column reg left parenthesis upper I left parenthesis script upper H 2 right parenthesis Superscript left bracket k minus a 1 right bracket Baseline right parenthesis greater than or equals left parenthesis StartAbsoluteValue upper V left parenthesis upper M Superscript prime Baseline right parenthesis EndAbsoluteValue minus StartAbsoluteValue upper M Superscript prime Baseline EndAbsoluteValue right parenthesis plus left parenthesis k minus a 1 right parenthesis 2nd Column equals left parenthesis a minus StartAbsoluteValue upper V left parenthesis upper M 1 right parenthesis EndAbsoluteValue plus a 1 right parenthesis plus left parenthesis k minus a 1 right parenthesis 2nd Row 1st Column Blank 2nd Column equals a minus StartAbsoluteValue upper V left parenthesis upper M 1 right parenthesis EndAbsoluteValue plus k period EndLayout

Thus, by (3), we have

$$\begin{align*}\mathrm{reg} \left( I(\mathcal{H}[V(M)])^{[k]} \right) \geq |V(M_1)| + \mathrm{reg}\left( I(\mathcal{H}_2)^{[k-a_1]} \right) \geq |V(M_1)| + \left(a -|V(M_1)|+ k\right)= a+k, \end{align*}$$

as desired. Now we can assume that $|M|=r+k-1$ . We have

$$\begin{align*}|M'| = \sum_{i=2}^r a_i = |M|-a_1 = (r+k-1)-a_1 = (r-1)+ (k-a_1+1)-1. \end{align*}$$

Since $r\geq 2$ , we have

$$\begin{align*}k-a_1+1\leq (r-2) + (k-a_1+1) = |M'| = (r-1)+ (k-a_1+1)-1. \end{align*}$$

Thus, $M'=\sqcup _{i=2}^r M_i$ is a generalized $(k-a_1+1)$ -admissible matching in $\mathcal {H}_2$ . By induction, we have

$$ \begin{align*} \mathrm{reg}\left( I(\mathcal{H}_2)^{[k-a_1+1]} \right) \geq (|V(M')| -|M'|) + (k-a_1+1) &= (a-|V(M_1)| + a_1) + (k-a_1+1) \\ &= a-|V(M_1)|+k+1. \end{align*} $$

Thus, by (3), we have

$$ \begin{align*} \mathrm{reg} \left( I(\mathcal{H}[V(M)])^{[k]} \right) &\geq |V(M_1)| + \mathrm{reg}\left( I(\mathcal{H}_2)^{[k-a_1+1]} \right) -1\\ &\geq |V(M_1)| + \left(a-|V(M_1)|+k+1 \right)-1 \\ &= a+k, \end{align*} $$

StartLayout 1st Row 1st Column reg left parenthesis upper I left parenthesis script upper H left bracket upper V left parenthesis upper M right parenthesis right bracket right parenthesis Superscript left bracket k right bracket Baseline right parenthesis 2nd Column greater than or equals StartAbsoluteValue upper V left parenthesis upper M 1 right parenthesis EndAbsoluteValue plus reg left parenthesis upper I left parenthesis script upper H 2 right parenthesis Superscript left bracket k minus a 1 plus 1 right bracket Baseline right parenthesis minus 1 2nd Row 1st Column Blank 2nd Column greater than or equals StartAbsoluteValue upper V left parenthesis upper M 1 right parenthesis EndAbsoluteValue plus left parenthesis a minus StartAbsoluteValue upper V left parenthesis upper M 1 right parenthesis EndAbsoluteValue plus k plus 1 right parenthesis minus 1 3rd Row 1st Column Blank 2nd Column equals a plus k comma EndLayout

as desired. This concludes the proof.

In particular, the lower bound in this result is sharp, as equality is achieved in the case where $I(\mathcal {H})$ is a complete intersection (Proposition 3.10).

This lower bound can be related to more familiar concepts in the literature in the case of d-uniform hypergraphs and, in particular, simple graphs.

Definition 3.12. Let $\mathcal {H}$ be a d-uniform hypergraph. For any $1\leq k\leq \nu (\mathcal {H})$ , we say that a matching M of $\mathcal {H}$ is a k-admissible matching if there is a partition $M=M_1\sqcup \cdots \sqcup M_r$ such that

1. for any edge $\mathcal {E} \text { in } \mathcal {H}[V(M)]$ , we have $\mathcal {E}\in E(\mathcal {H}[V(M_i)])$ for some $i\in [r]$ ;
2. $|M|\leq r+k-1$ .

Now, we define the k-admissible matching number of $\mathcal {H}$ , denoted by $\mathrm {aim}(\mathcal {H}, k)$ , as follows $:$

$$\begin{align*}\mathrm{aim}(\mathcal{H}, k) := \max\{|M| \mid M \text{ is a } k\text{-admissible matching of } \mathcal{H}\}. \end{align*}$$

We note that our definition of k-admissible matching is a refinement of [Reference Erey and Hibi15, Definition 12] in the case when $\mathcal {H}$ is a simple graph G. Observe that for a simple graph G, condition (1) in Definition 3.12 is the same as saying that the edges $e_i\in M_i$ and $e_j\in M_j$ form a gap (in the sense of Erey and Hibi [Reference Erey and Hibi15]) for any $i\neq j$ . The only difference, therefore, is that Erey and Hibi considered an additional assumption, namely, the induced subgraphs on $M_i$ for each $i\in [r]$ need to be forests.

The following facts are straightforward to verify; interested readers may look at the analogous results in [Reference Erey and Hibi15, p. 6].

Remark 3.13. Let $\mathcal {H}$ be a d-uniform hypergraph.

1. A matching M of the hypergraph $\mathcal {H}$ is a $1$ -admissible matching if and only if M is an induced matching of $\mathcal {H}$ . Thus, $\nu _1(\mathcal {H}) = \mathrm {aim}(\mathcal {H}, 1)$ .
2. If $1\leq k \leq \nu (\mathcal {H})$ , then every k-admissible matching of $\mathcal {H}$ is also a $(k+1)$ -admissible matching. In particular,
$$\begin{align*}\nu_1(\mathcal{H})=\mathrm{aim}(\mathcal{H},1)\leq \mathrm{aim}(\mathcal{H},2)\leq \cdots \leq \mathrm{aim}(\mathcal{H},\nu(\mathcal{H}))\leq \nu(\mathcal{H}). \end{align*}$$
3. For any $1\leq k\leq \nu (\mathcal {H})$ , $\mathrm {aim}(\mathcal {H}, k)\geq k$ .
4. For any $2\leq k\leq \nu (\mathcal {H})$ , $\mathrm {aim}(\mathcal {H}, k)\leq \mathrm {aim}(\mathcal {H}, k-1)+1$ .

Remark 3.14. Let $\mathcal {H}$ be a d-uniform hypergraph. Then, one can verify that any k-admissible matching M of $\mathcal {H}$ with $\vert M\vert \geq k$ is actually a generalized k-admissible matching. For the computation of $\mathrm {aim}(\mathcal {H},k)$ , the following proposition suggests that it is enough to consider only generalized k-admissible matching of $\mathcal {H}$ .

Proposition 3.15. Let $\mathcal {H}$ be a d-uniform hypergraph and $1\leq k\leq \nu (\mathcal {H})$ . Then

$$\begin{align*}(d-1)\mathrm{aim}(\mathcal{H},k) = \max\{ |V(M)|-|M|\colon M \text{ is a generalized } k\text{-admissible matching of } \mathcal{H}\}. \end{align*}$$

Proof. Note that if M is a generalized k-admissible matching of M, then M is also a k-admissible matching, and $|V(M)|-|M|=(d-1)|M|$ . Therefore, by Remark 3.14, to prove the equality, it is enough to show that if M is a k-admissible matching of $\mathcal {H}$ such that $|M|=\mathrm {aim}(\mathcal {H},k)$ , then $|M|\ge k$ . Evidently, since $k\le \nu (\mathcal {H})$ , there exists a matching $M'$ of $\mathcal {H}$ such that $|M'|=k$ . Thus, if $|M|<k$ , then $|M|<|M'|$ , where $M'=M_1'$ is a k-admissible matching of $\mathcal {H}$ (by Definition 3.12), a contradiction.

A straightforward and important consequence of our previous results is that we can use $\mathrm {aim}(\mathcal {H},k)$ to establish a lower bound for the regularity of square-free powers of $I(\mathcal {H})$ , in the case $\mathcal {H}$ is a d-uniform hypergraph.

Theorem 3.16. Let $\mathcal {H}$ be a d-uniform hypergraph and k an integer where $1\leq k\leq \nu (\mathcal {H})$ . Then

$$\begin{align*}\mathrm{reg} \left( I(\mathcal{H})^{[k]} \right) \geq (d-1)\mathrm{aim}(\mathcal{H},k) +k. \end{align*}$$

Specifically, for a simple graph G and for any $1\leq k\leq \nu (G)$ , we have

$$\begin{align*}\mathrm{reg} \left( I(G)^{[k]}\right) \geq \mathrm{aim}(G,k)+k. \end{align*}$$

Proof. The result follows immediately from Theorem 3.11 and Proposition 3.15.

Remark 3.17. To distinguish the notion of k-admissible matching by Erey and Hibi from the one defined here, we denote their k-admissible matching number by $\mathrm {aim}^*(G,k)$ (In [Reference Erey and Hibi15, Definition 13], they denoted this number by $\mathrm {aim}(G,k)$ ). For a simple graph G, it is immediate from the two definitions that $\mathrm {aim}^*(G,k)\leq \mathrm {aim}(G,k)$ . Note that $\mathrm {aim}^*(G,k)=\mathrm {aim}(G,k)$ , when G is a forest. However, the difference can be arbitrarily large in the case of arbitrary graphs. For instance, if we consider G to be the complete graph on $2k$ vertices, then $\mathrm {aim}(G,k)=k$ , whereas $\mathrm {aim}^*(G,k)=1$ .

4. Square-free powers of block graphs

In this section, we consider the family of block graphs and derive a combinatorial formula for the regularity of their square-free powers in terms of the admissible matching numbers. The class of block graphs is a natural generalization of forests. Thus, our formula can be seen as an extension of the regularity formula proved in [Reference Crupi, Ficarra and Lax6, Theorem 3.6] for the square-free powers of forests.

A block graph G is a chordal graph such that any two maximal cliques in G intersect in at most one common vertex. Before going to the proof of the regularity formula, we first introduce some notations and terminologies related to block graphs. Based on these, we are able to establish some structural properties of block graphs that will be helpful later in this section.

Definition 4.1. Let G be a block graph.

• A maximal clique in G is called a block. Let $\mathcal {B}_G$ denote the set of all blocks of G.
• A block $B'\in \mathcal {B}_G$ is said to be a block neighbor of $B\in \mathcal {B}_G$ if $|V(B)\cap V(B')|=1$ . The set $\{B'\in \mathcal {B}_G\colon |V(B)\cap V(B')|=1\}$ is called the block neighborhood of B in G and is denoted by $\mathcal {N}_G(B)$ .
• A block $B\in \mathcal {B}_G$ is said to be a leaf block if the number of free vertices in B is at least $|V(B)|-1$ .
• A block $B\in \mathcal {B}_G$ is said to be a distant leaf block if B is a leaf block and at most one block neighbor of B in G is not a leaf block.
• Let G be a block graph, and $B_1,B_2$ two different blocks in G. A block path between $B_1$ and $B_2$ of length n is a collection of blocks $\mathcal {P}: D_1,D_2,\ldots , D_n$ satisfying the following conditions:

(i) $D_1=B_1$ , $D_n=B_2$ ;
(ii) $D_i\neq D_j$ for each $i\neq j$ ;
(iii) for each $i\neq j$ , $V(D_i)\cap V(D_j)\neq \emptyset $ if and only if $j=i-1$ or $j=i+1$ .

We explain the above terminologies in the following example.

Example 4.2. Let G be the block graph as in Figure 1. The block on the vertex set $\{x_{13},x_{14}\}$ is a distant leaf block, and its block neighbors are the blocks on the vertex sets $\{x_{13},x_{15}\}$ and $\{x_{10},x_{11},x_{12},x_{13}\}$ . Note that the block on the vertex set $\{x_6,x_9\}$ is a leaf block but not a distant leaf block. The block path $\mathcal {P}:D_1,D_2,D_3$ is the (unique) block path between $D_1$ and $D_3$ , where $V(D_1)=\{x_1,x_2,x_3\}$ , $V(D_2)=\{x_3,x_4,x_5,x_6\},$ and $V(D_3)=\{x_6,x_{10}\}$ .

Figure 1

A block graph G.

Figure 1 Long description

The graph is composed of several interconnected components and one isolated edge.

Starting from the left, vertices x sub 1, x sub 2, and x sub 3 form a triangle. Vertex x sub 3 acts as a bridge to the next cluster.

The second cluster is a diamond shape formed by vertices x sub 3, x sub 4, x sub 5, and x sub 6. All four vertices are interconnected, with a vertical edge between x sub 4 and x sub 5 and a horizontal edge between x sub 3 and x sub 6.

Vertex x sub 6 is a central hub. It is connected to a triangle formed by x sub 7 and x sub 8, and it has a single pendant edge leading to x sub 9.

A horizontal edge connects x sub 6 to x sub 10. Vertex x sub 10 is part of another diamond cluster with x sub 11, x sub 12, and x sub 13. This cluster mirrors the previous one, with a vertical edge between x sub 11 and x sub 12 and a horizontal edge between x sub 10 and x sub 13.

Vertex x sub 13 has two pendant edges branching out to the right, leading to vertices x sub 14 and x sub 15.

Separated from the main structure on the far right is an isolated edge connecting vertex x sub 16 to vertex x sub 17.

We remark that if G is a tree, then a distant leaf block is just a distant leaf in the sense of Crupi et al. [Reference Crupi, Ficarra and Lax6]. Recall that in a tree, there is a unique path between any two distinct vertices. Analogously, in a connected block graph G, one can show that there is a unique block path between any two distinct blocks. Consequently, every block graph contains a distant leaf block, and we state this fact formally as a lemma below.

Lemma 4.3. Let G be a block graph. Then, G contains a distant leaf block.

The following lemma offers a useful criterion for identifying distant leaf blocks in an arbitrary block graph.

Lemma 4.4. Let $\mathcal {P}:D_1,D_2,\ldots ,D_n$ be a block path in a block graph G of the maximum possible length. Then the following is true $:$

(1) $D_1$ and $D_n$ are distant leaf blocks of G.
(2) If $B\in \mathcal {N}_G(D_2)\setminus \{D_3\}$ such that $V(B)\cap V(D_2)\neq V(D_2)\cap V(D_3)$ , then B is a distant leaf block in G.

Proof. (1) is easy to verify since G does not contain any induced cycle of length $\geq 4$ . For (2), suppose $V(D_i)\cap V(D_{i+1})=\{x_i\}$ for each $i\ge 2$ and $V(B)\cap V(D_2)=\{z\}$ for some $z\in V(G)$ . By the given hypothesis, we have $z\neq x_2$ . Observe that $V(B)\cap V(D_i)=\emptyset $ for each $i>2$ . Hence, $\mathcal {P}':B,D_2,\ldots ,D_n$ is a block path of the maximum possible length in G. Therefore, from (1), we see that B is a distant leaf block in G.

Next, we derive some results on the k-admissible matching number of block graphs. Although analogous statements hold for arbitrary graphs, we state and prove them here only in the context of block graphs.

Proposition 4.5. Let G be a block graph and $B\in \mathcal {B}_G$ . Let $\{u_1,\ldots ,u_l\}\subseteq V(B)$ for some $l\ge 1$ , and H denote the induced subgraph $G\setminus \{u_1,\ldots ,u_l\}$ . Then for each $2\le k\le \nu (G)$ , we have $\mathrm {aim}(H,k-1)\le \mathrm {aim}(G,k)-1$ if one of the following holds:

(i) For some $i\in [l]$ , $|\mathcal {N}_G[u_i]\setminus V(B)|\ge 1$ , and for each $v\in \mathcal {N}_G[u_i]\setminus V(B)$ , $\deg (v)=1$ .
(ii) $l\ge 2$ , and $u_1,u_2$ are two distinct free vertices in G.

Proof. First, let us assume that condition (i) holds. Let $\mathcal {N}_G[u_i]\setminus V(B)=\{v_1,\ldots ,v_r\}$ for some $r\ge 1$ . Let $M=\sqcup _{i=1}^rM_i$ be a $(k-1)$ -admissible matching of H such that $|M|=\mathrm {aim}(H,k-1)$ . Since B is a block in G, we see that $n:= |\{i\mid V(M_i)\cap V(B)\neq \emptyset \}|\le 1$ . If $n=0$ , then it is easy to see that $M'=\sqcup _{i=1}^{r+1}M_i'$ , where $M_i'=M_i$ for each $i\in [r]$ , and $M_{r+1}'=\{\{u_i,v_1\}\}$ , forms a k-admissible matching of G. Now, suppose $n=1$ and without loss of generality, let $V(M_r)\cap V(B)\neq \emptyset $ . In this case, $M'=\sqcup _{i=1}^{r}M_i'$ , where $M_i'=M_i$ for each $i\in [r-1]$ , and $M_r'=M_r\cup \{\{u_i,v_1\}\}$ , forms a k-admissible matching of G. Thus, combining both cases, we obtain $\mathrm {aim}(H,k-1)\le \mathrm {aim}(G,k)-1$ .

For (ii), working with the edge $\{u_1,u_2\}$ similar to $\{u_i,v_1\}$ above and proceeding accordingly, we also obtain $\mathrm {aim}(H,k-1)\le \mathrm {aim}(G,k)-1$ . This completes the proof.

Proposition 4.6. Let G be a block graph and $B\in \mathcal {B}_G$ such that $u_1,u_2\in V(B)$ are two distinct free vertices in G. If $H=G\setminus V(B)$ , then for each $1\le k\le \nu (G)$ , we have $\mathrm {aim}(H,k)\le \mathrm {aim}(G,k)-1$ .

Proof. Let $M=\sqcup _{i=1}^rM_i$ be a k-admissible matching of H such that $|M|=\mathrm {aim}(H,k)$ . Then $M'=\sqcup _{i=1}^{r+1}M_i'$ , where $M_i'=M_i$ for each $i\in [r]$ , and $M_{r+1}'=\{\{u_1,u_2\}\}$ , forms a k-admissible matching of G. Thus, $\mathrm {aim}(H,k)\le \mathrm {aim}(G,k)-1$ .

Now, we will introduce the notion of a special block in a block graph, which plays a crucial role in proving our main theorem in this section. Let G be a block graph with $B\in \mathcal {B}_G$ and $u\in V(B)$ . Define $\mathcal {N}_G(B,u):= \{D\in \mathcal {B}_G\mid V(D)\cap V(B)=\{u\}\}$ .

Definition 4.7. Let G be a block graph, and B a block of G with $V(B)=\{u_1,\ldots ,u_d\}$ . We say that B is a special block of G if B is one of the following types $:$

Type I. $d\le 2$ , and $\mathcal {N}_G(B,u_i)=\emptyset $ for each $i\in [d]$ .
Type II. $d\ge 3$ , and $\mathcal {N}_G(B,u_i)=\emptyset $ for each $i\in [d-1]$ .
Type III. $d\ge 2$ , and there exists some $i\in [d-1]$ such that $\mathcal {N}_G(B,u_i)\neq \emptyset $ . Moreover, if $\mathcal {N}_G(B,u_i)\neq \emptyset $ for some $i\in [d-1]$ , then for each $D\in \mathcal {N}_G(B,u_i)$ , $D\cong K_2$ .

Example 4.8. Let G be the block graph as in Figure 1. Then the block on the vertex set $\{x_{16},x_{17}\}$ is a special block of Type I, whereas the block on the vertex set $\{x_1,x_2,x_3\}$ is a special block of Type II. On the other hand, the block on the vertex set $\{x_{10},x_{11},x_{12},x_{13}\}$ is a special block of Type III.

Lemma 4.9. Let G be a block graph. Then, G contains at least one special block.

Proof. First, consider the case when G contains an isolated vertex $\{u\}$ . In this case, $B=G[\{u\}]$ is a special block of Type I in G. Next, suppose G has no isolated vertices. Let

$$\begin{align*}\Sigma=\{B\in \mathcal{B}_G\mid \mathcal{P}:B=D_1,D_2,\ldots,D_n\text{ is a block path of maximum possible length in }G\}. \end{align*}$$

Then, it is easy to see that B is a distant leaf block of G. Observe that for each $B\in \Sigma $ , $|V(B)|\ge 2$ . If $|V(B)|\ge 3$ , for some $B\in \Sigma $ , then B is a special block of Type II in G.

Next, suppose for each $B\in \Sigma $ , we have $|V(B)|=2$ . Fix some $B\in \Sigma $ and let $\mathcal {P}:B=D_1,D_2,\ldots ,D_n$ be a block path of the maximum possible length in G. If $n=1$ , then G is a disjoint union of edges, and thus B is a special block of Type I in G.

Now consider the case when $n\ge 2$ , that is, $\mathcal {P}$ is a block path of length at least $2$ . Let $V(D_2)=\{u_1,\ldots ,u_d\}$ , where $V(D_1)\cap V(D_2)=\{u_1\}$ and $V(D_2)\cap V(D_3)=\{u_d\}$ . Here, we remark that the value of n can also be $2$ , and in that case, no such $D_3$ exists. Without loss of generality, let $\mathcal {N}_G(D_2)=\{\mathcal {E}_{i,j}\mid i\in [d],j\le r_i\}$ , where $r_i$ is a non-negative integer for each $i\in [d]$ , $V(\mathcal {E}_{i,j})\cap V(D_2)=\{u_i\}$ , $\mathcal {E}_{1,1}=D_1$ , and $\mathcal {E}_{d,1}=D_3$ . If $r_i=0$ for some $i\in [d]$ , then we make the convention that $u_i$ is a free vertex in G so that no such $\mathcal {E}_{i,j}$ exists. Thus, we always have $r_1\ge 1$ . Since $\mathcal {P}$ is a block path of maximum length in G, by Lemma 4.4, we have that $\mathcal {E}_{i,j}$ is a distant leaf block in G for each $i\in [d-1]$ and $j\in [r_i]$ . Moreover, $\mathcal {E}_{i,j}\in \Sigma $ for such i and j. Then by our assumption $|V(\mathcal {E}_{i,j})|=2$ for each $i\in [d-1]$ and $j\in [r_i]$ , that is, $\mathcal {E}_{i,j}\cong K_2$ . Consequently, $D_2$ is a special block of Type III in G, and this completes the proof.

Notation 4.10. Let B be a special block in a block graph G. Let $V(B)=\{u_1,\ldots ,u_d\}$ , and for each $i\in [d-1]$ , either $\mathcal {N}_G(B,u_i)=\emptyset $ or if for some $i\in [d-1]$ , $\mathcal {N}_G(B,u_i)\neq \emptyset $ , then for each $D\in \mathcal {N}_G(B,u_i)$ , we have $D\cong K_2$ . Define

$$\begin{align*}\Lambda:=\cup_{i=1}^{d-1}\mathcal{N}_G(B,u_i). \end{align*}$$

Then $\Lambda $ is a collection of blocks in G which are isomorphic to $K_2$ . Now, for each $S\subseteq \Lambda $ , we define the ideal

$$\begin{align*}I_{S,B}=\langle xy\mid x,y\in V(D) \text{ for some }D\in S\rangle. \end{align*}$$

Theorem 4.11. Let G be a block graph on n vertices. Then, for each $1\le k\le \nu (G)$ ,

(i) $\mathrm {reg}(I(G)^{[k]})\le \mathrm {aim}(G,k)+k$ .
(ii) $\mathrm {reg}(I(G)^{[k]}+I_{S,B})\le \mathrm {aim}(G,k)+k$ , where B is a special block in G, and $I_{S,B}$ is the ideal associated with any $S\subseteq \Lambda $ as described in Notation 4.10.

Consequently, for a block graph G, we have $\mathrm {reg}(I(G)^{[k]})=\mathrm {aim}(G,k)+k$ .

Proof. We prove (1) and (2) simultaneously by induction on n. By Lemma 4.9, G contains at least one special block. Let B be a special block in G, and $I_{S,B}$ be the ideal defined as in Notation 4.10, where $S\subseteq \Lambda $ . Now, if $n\le 3$ , then $k=1$ , and thus it is easy to see that

$$\begin{align*}\mathrm{reg}(I(G)^{[1]}+I_{S,B})=\mathrm{reg}(I(G)^{[1]})=\nu_1(G)+1=\mathrm{aim}(G,1)+1. \end{align*}$$

Therefore, we may assume that $n\ge 4$ . For such an n, the case $k=1$ follows from [Reference Ha and Van Tuyl20, Theorem 6.8] since $I(G)^{[1]}+I_{S,B}=I(G)^{[1]}$ and $\nu _1(G)=\mathrm {aim}(G,1)$ . Therefore, we can assume $k\ge 2$ .

Case I: B is a special block of Type I in G. In this case, B is a connected component of G, and either B is an isolated vertex in G or $B\cong K_2$ . In both cases, we see that $\Lambda =\emptyset $ and $I_{S,B}=\langle 0\rangle $ . Now if B is an isolated vertex of G, then by the induction hypothesis and Proposition 3.3, we have

$$ \begin{align*} \mathrm{reg}(I(G)^{[k]})&=\mathrm{reg}(I(G\setminus V(B))^{[k]})\\ &\le \mathrm{aim}(G\setminus V(B),k)+k\\ &\le\mathrm{aim}(G,k)+k. \end{align*} $$

StartLayout 1st Row 1st Column reg left parenthesis upper I left parenthesis upper G right parenthesis Superscript left bracket k right bracket Baseline right parenthesis 2nd Column equals reg left parenthesis upper I left parenthesis upper G minus upper V left parenthesis upper B right parenthesis right parenthesis Superscript left bracket k right bracket Baseline right parenthesis 2nd Row 1st Column Blank 2nd Column less than or equals aim left parenthesis upper G minus upper V left parenthesis upper B right parenthesis comma k right parenthesis plus k 3rd Row 1st Column Blank 2nd Column less than or equals aim left parenthesis upper G comma k right parenthesis plus k period EndLayout

Now, suppose $B\cong K_2$ and $V(B)=\{u,v\}$ . Then,

(4) $$ \begin{align} \mathrm{reg}((I(G)^{[k]}+\langle uv\rangle):u)&=\mathrm{reg}(I(G\setminus u)^{[k]}+\langle v\rangle) \nonumber\\ &\le \mathrm{aim}(G\setminus u,k)+k\nonumber\\ &\le\mathrm{aim}(G,k)+k-1, \end{align} $$

StartLayout 1st Row 1st Column reg left parenthesis left parenthesis upper I left parenthesis upper G right parenthesis Superscript left bracket k right bracket Baseline plus left angle bracket u v right angle bracket right parenthesis colon u right parenthesis 2nd Column equals reg left parenthesis upper I left parenthesis upper G minus u right parenthesis Superscript left bracket k right bracket Baseline plus left angle bracket v right angle bracket right parenthesis 2nd Row 1st Column Blank 2nd Column less than or equals aim left parenthesis upper G minus u comma k right parenthesis plus k 3rd Row 1st Column Blank 2nd Column less than or equals aim left parenthesis upper G comma k right parenthesis plus k minus 1 comma EndLayout

where the equality follows from [Reference Das, Roy and Saha8, Lemma 2.9], and the first and second inequalities follow from the induction hypothesis and Proposition 4.6, respectively. Next,

(5) $$ \begin{align} \mathrm{reg}(I(G)^{[k]}+\langle uv\rangle+\langle u\rangle)&=\mathrm{reg}((I(G\setminus u)^{[k]}+\langle u\rangle)\nonumber\\ &\le\mathrm{aim}(G\setminus u,k)+k\nonumber\\ &\le \mathrm{aim}(G,k)+k, \end{align} $$

where, as before, the equality follows from [Reference Das, Roy and Saha8, Lemma 2.9], and the first and second inequalities follow from the induction hypothesis and Proposition 4.6, respectively. Now, using inequalities (4), (5), and Lemma 2.1 together, we obtain

(6) $$ \begin{align} \mathrm{reg}(I(G)^{[k]}+\langle uv\rangle)\le\mathrm{aim}(G,k)+k. \end{align} $$

Next, by [Reference Das, Roy and Saha8, Lemma 2.9], $(I(G)^{[k]}:uv)=I(G\setminus \{u,v\})^{[k-1]}$ . Observe that both u and v are free vertices of G. Therefore, using the induction hypothesis and Proposition 4.5, we obtain

(7) $$ \begin{align} \mathrm{reg}(I(G)^{[k]}:uv)&=\mathrm{reg}(I(G\setminus\{u,v\})^{[k-1]})\nonumber\\ &\le\mathrm{aim}(G\setminus\{u,v\},k-1)+k-1\nonumber\\ &\le\mathrm{aim}(G,k)+k-2. \end{align} $$

Combining inequalities (6) and (7) and using Lemma 2.1, we get $\mathrm {reg}(I(G)^{[k]})\le \mathrm {aim}(G,k)+k$ . This completes the proof of (ii) as well as (i).

Case II: B is a special block of Type II in G. Hence, $|V(B)|\ge 3$ . As in Case I, in this case too, we have $\Lambda =\emptyset $ and $I_{S,B}=\langle 0\rangle $ . Let $V(B)=\{u_1,\ldots ,u_d\}$ , where $d\ge 3$ , and $u_1,\ldots ,u_{d-1}$ are free vertices in G.

Claim 1. For each $j\in [d-1]$ ,

$$\begin{align*}\mathrm{reg}((I(G)^{[k]}+\langle u_1u_2,\ldots,u_1u_j\rangle):u_1u_{j+1})\le \mathrm{aim}(G,k)+k-2, \end{align*}$$

where $(I(G)^{[k]}+\langle u_1u_2,\ldots ,u_1u_j\rangle =I(G)^{[k]}$ in case $j=1$ .

Proof of Claim 1.

Let $G_j=G\setminus \{u_1,\ldots ,u_{j+1}\}$ . Then,

$$ \begin{align*} \mathrm{reg}((I(G)^{[k]}+\langle u_1u_2,\ldots,u_1u_j\rangle):u_1u_{j+1})&=\mathrm{reg}(I(G_j)^{[k-1]}+\langle u_2,\ldots,u_j\rangle)\\ &\le \mathrm{aim}(G_j,k-1)+k-1\\ &\le \mathrm{aim}(G,k)+k-2, \end{align*} $$

where the first equality follows from [Reference Das, Roy and Saha8, Lemma 2.9], and the first and second inequalities follow from the induction hypothesis and Proposition 4.5, respectively. This completes the proof of Claim 1.

Let $H=G\setminus N_G[u_1]$ . Then,

(8) $$ \begin{align} \mathrm{reg}((I(G)^{[k]}+\langle u_1u_2,\ldots,u_1u_d\rangle):u_1)&=\mathrm{reg}(I(H)^{[k]}+\langle u_2,\ldots,u_d\rangle)\nonumber\\ &\le \mathrm{aim}(H,k)+k\nonumber\\ &\le \mathrm{aim}(G,k)+k-1, \end{align} $$

where the first equality again follows from [Reference Das, Roy and Saha8, Lemma 2.9], and the first and second inequalities follow from the induction hypothesis and Proposition 4.6, respectively. Next, we have

(9) $$ \begin{align} \mathrm{reg}(I(G)^{[k]}+\langle u_1u_2,\ldots,u_1u_d,u_1\rangle)&=\mathrm{reg}(I(G\setminus u_1)^{[k]})\nonumber\\ &\le\mathrm{aim}(G\setminus u_1,k)+k\nonumber\\ &\le \mathrm{aim}(G,k)+k, \end{align} $$

where the first and second inequalities follow from the induction hypothesis and Proposition 3.3, respectively.

Now, by Lemma 2.1 along with inequalities (8) and (9), we get $\mathrm {reg}(I(G)^{[k]}+\langle u_1u_2,\ldots ,u_1u_d\rangle ) \le \mathrm {aim}(G,k)+k$ . Observe that, from Claim 1, we have $\mathrm {reg}((I(G)^{[k]}+\langle u_1u_2,\ldots ,u_1u_{d-1}\rangle ):u_1u_d)\le \mathrm {aim}(G,k)+k-2$ , and hence, using Lemma 2.1 again, we obtain $\mathrm {reg}(I(G)^{[k]}+\langle u_1u_2,\ldots ,u_1u_{d-1}\rangle )\le \mathrm {aim}(G,k)+k$ . By a repeated use of Claim 1 and Lemma 2.1, we finally get $\mathrm {reg}(I(G)^{[k]})\le \mathrm {aim}(G,k)+k$ , as desired.

Case III: B is a special block of Type III in G. Let $V(B)=\{u_1,\ldots ,u_d\}$ , and without loss of generality, let $\mathcal {N}_G(B,u_1)\neq \emptyset $ . Now, for each $i\in [d-1]$ , let $\mathcal {N}_G(B,u_i)=\{\mathcal {E}_{i,j}\mid j\le r_i\}$ , where $r_i$ is a non-negative integer. As mentioned in the proof of Lemma 4.9, if $r_i=0$ for some $i\in [d]$ , then we make the convention that $u_i$ is a free vertex in G. Thus, we always have $r_1\ge 1$ . By our assumption, we also have $\mathcal {E}_{i,j}\cong K_2$ . If such an $\mathcal {E}_{i,j}$ exists, let $V(\mathcal {E}_{i,j})=\{u_i,v_{i,j}\}$ for each $i\in [d-1]$ and $j\in [r_i]$ . In this case, our first claim is the following.

Claim 2. $\mathrm {reg}(I(G)^{[k]}+I_{\Lambda ,B})\le \mathrm {aim}(G,k)+k$ .

Proof of Claim 2.

By [Reference Das, Roy and Saha8, Lemma 2.9], for each $p\in [d-1]$ , we have

(10) $$ \begin{align} &((I(G)^{[k]}+ I_{\Lambda,B}+\langle u_1u_2,\ldots, u_1u_p\rangle):u_1u_{p+1}) \nonumber\\ &=I(G\setminus \{u_1,\ldots,u_p,u_{p+1}\})^{[k-1]}+\langle u_iv_{i,j}\mid p+2\le i\le d-1\rangle\nonumber\\ &\hspace{4cm}+\langle v_{1,j},v_{p+1,l}\mid j\in [r_1],p+1\le d-1, l\in[r_{p+1}]\rangle. \end{align} $$

StartLayout 1st Row 1st Column Blank 2nd Column left parenthesis left parenthesis upper I left parenthesis upper G right parenthesis Superscript left bracket k right bracket Baseline plus upper I Subscript normal upper Lamda comma upper B Baseline plus left angle bracket u 1 u 2 comma ellipsis comma u 1 u Subscript p Baseline right angle bracket right parenthesis colon u 1 u Subscript p plus 1 Baseline right parenthesis 2nd Row 1st Column Blank 2nd Column equals upper I left parenthesis upper G minus StartSet u 1 comma ellipsis comma u Subscript p Baseline comma u Subscript p plus 1 Baseline EndSet right parenthesis Superscript left bracket k minus 1 right bracket Baseline plus left angle bracket u Subscript i Baseline v Subscript i comma j Baseline vertical bar p plus 2 less than or equals i less than or equals d minus 1 right angle bracket 3rd Row 1st Column Blank 2nd Column plus left angle bracket v Subscript 1 comma j Baseline comma v Subscript p plus 1 comma l Baseline vertical bar j element of left bracket r 1 right bracket comma p plus 1 less than or equals d minus 1 comma l element of left bracket r Subscript p plus 1 Baseline right bracket right angle bracket period EndLayout

Here, we make the convention that $I(G)^{[k]}+I_{\Lambda ,B}+\langle u_1u_2,\ldots , u_1u_p\rangle =I(G)^{[k]}+I_{\Lambda ,B}$ , in case $p=1$ . Note that, in Equation (10), we have $\langle u_iv_{i,j}\mid p+2\le i\le d-1\rangle =\langle 0\rangle $ if $p\in \{d-2,d-1\}$ . Moreover, if $p=d-3$ and $r_{d-1}=0$ , then we again have $\langle u_iv_{i,j}\mid p+2\le i\le d-1\rangle =\langle 0\rangle $ . In all other cases, let $B'=B\setminus \{u_1,\ldots ,u_{p+1}\}$ . Then it is easy to see that $B'$ is a special block in $G\setminus \{u_1,\ldots ,u_{p+1}\}$ . Moreover, $I_{\Lambda ',B'}=\langle u_iv_{i,j}\mid p+2\le i\le d-1\rangle $ , where $\Lambda '=\cup _{i=p+2}^{d-1}\mathcal {N}_{G\setminus \{u_1,\ldots ,u_p,u_{p+1}\}}(B',u_i)$ . Now, by the induction hypothesis and using Proposition 4.5, we obtain

(11) $$ \begin{align} \begin{aligned} \mathrm{reg}((I(G)^{[k]}+I_{\Lambda,B}+\langle u_1u_2,\ldots, u_1u_p\rangle):u_1u_{p+1})&\le \mathrm{aim}(G\setminus \{u_1,\ldots,u_{p+1}\},k-1)+k-1\\ &\le \mathrm{aim}(G,k)+k-2. \end{aligned} \end{align} $$

In particular,

(12) $$ \begin{align} \mathrm{reg}((I(G)^{[k]}+I_{\Lambda,B}+\langle u_1u_2,\ldots, u_1u_{d-1}\rangle):u_1u_{d})\le \mathrm{aim}(G,k)+k-2. \end{align} $$

Next, we have

(13) $$ \begin{align} \mathrm{reg}((I(G)^{[k]}+I_{\Lambda,B}+\langle u_1u_2,\ldots, u_1u_{d}\rangle):u_1)&=\mathrm{reg}(I(G\setminus \{u_1,\ldots,u_d\})^{[k]})\nonumber\\ &\le \mathrm{aim}(G\setminus \{u_1,\ldots,u_d\},k)+k\nonumber\\ &\le \mathrm{aim}(G,k)+k-1, \end{align} $$

where the equality follows from [Reference Das, Roy and Saha8, Lemma 2.9], and the first and second inequalities follow from the induction hypothesis and Proposition 4.6, respectively. Moreover, $I(G)^{[k]}+I_{\Lambda ,B}+\langle u_1u_2,\ldots , u_1u_{d}\rangle +\langle u_1\rangle =I(G\setminus u_1)^{[k]}+\langle u_iv_{i,j}\mid 2\le i\le d-1, j\in [r_i]\rangle $ . As before, the ideal $\langle u_iv_{i,j}\mid 2\le i\le d-1, j\in [r_i]\rangle =\langle 0\rangle $ if $d=2$ or $d=3$ with $r_2=0$ . In all other cases, let $B"=B\setminus u_1$ . Then, it is easy to see that $B"$ is a special block in $G\setminus u_1$ . Moreover, $I_{\Lambda ",B"}=\langle u_iv_{i,j}\mid 2\le i\le d-1, j\in [r_i]\rangle $ , where $\Lambda "=\cup _{i=2}^{d-1}\mathcal {N}_{G\setminus u_1}(B",u_i)$ . Therefore, by the induction hypothesis and Proposition 3.3, we get

(14) $$ \begin{align} \begin{aligned} \mathrm{reg}(I(G)^{[k]}+I_{\Lambda,B}+\langle u_1u_2,\ldots, u_1u_{d}\rangle+\langle u_1\rangle)&\le \mathrm{aim}(G\setminus u_1,k)+k\\ &\le \mathrm{aim}(G,k)+k. \end{aligned} \end{align} $$

In view of Lemma 2.1, inequalities (13) and (14) give

$$ \begin{align*} \mathrm{reg}(I(G)^{[k]}+I_{\Lambda,B}+\langle u_1u_2,\ldots, u_1u_{d}\rangle\rangle)\le \mathrm{aim}(G,k)+k. \end{align*} $$

On the other hand, using inequality (12) and Lemma 2.1 again, we obtain

$$\begin{align*}\mathrm{reg}(I(G)^{[k]}+I_{\Lambda,B}+\langle u_1u_2,\ldots, u_1u_{d-1}\rangle\rangle)\le \mathrm{aim}(G,k)+k. \end{align*}$$

Thus, by a repeated use of inequality (11) and Lemma 2.1, we have $\mathrm {reg}(I(G)^{[k]}+I_{\Lambda , B})\le \mathrm {aim}(G,k)+k$ , and this completes the proof of Claim 2.

Claim 3. $\mathrm {reg}(I(G)^{[k]}+ I_{S,B})\le \mathrm {aim}(G,k)+k $ , where $S\subseteq \Lambda =\{u_iv_{i,j}\mid i\in [d-1]\text { and }j\in [r_i]\}$ .

Proof of Claim 4.

We induct on $|\Lambda |-|S|$ . When $S=\Lambda $ , it follows from Claim 2. Let $|S|<|\Lambda |$ , and choose some $u_lv_{l,t}\in \Lambda \setminus S$ . Then $l\in [d-1]$ and $t\in [r_l]$ . Let $S_1=S\cup \mathcal {E}_{l,t}$ . By the induction hypothesis, we have

(15) $$ \begin{align} \begin{aligned} \mathrm{reg}(I(G)^{[k]}+I_{S,B}+\langle u_lv_{l,t}\rangle)&=\mathrm{reg}(I(G)^{[k]}+I_{S_1,B})\\ &\le \mathrm{aim}(G,k)+k. \end{aligned} \end{align} $$

Now, by [Reference Das, Roy and Saha8, Lemma 2.9], $(I(G)^{[k]}+I_{S,B}): u_lv_{l,t})=I(\Gamma )^{[k-1]}+\langle u_iv_{i,j}\mid \mathcal {E}_{i,j}\in S\text { with }i\in [d-1]\setminus \{l\}\text { and }j\in [r_i]\rangle +\langle v_{l,q}\mid u_lv_{l,q}\in I_{S,B}\text { and }q\neq t\rangle $ , where $\Gamma =G\setminus \{u_l\}$ . Again, observe that if $d=2$ or $d=3$ with $\mathcal {E}_{i,j}\notin S$ for each $i\in [d-1]\setminus \{l\}$ and $j\in [r_i]$ , then the ideal $\langle u_iv_{i,j}\mid \mathcal {E}_{i,j} \in S\text { with }i\in [d-1]\setminus \{l\}\text { and }j\in [r_i]\rangle =\langle 0\rangle $ . In all other cases, let $B"'=B\setminus u_l$ . Then we see that $B"'$ is a special block in $G\setminus u_l$ . Moreover, $I_{S_1,B"'}=\langle u_iv_{i,j}\mid \mathcal {E}_{i,j}\in S\text { with }i\in [d-1]\setminus \{l\}\text { and }j\in [r_i]\rangle $ , where $S_1=S\cap (\cup _{\underset {i\neq l}{i=1}}^{d-1}\mathcal {N}_{G\setminus u_l}(B"',u_i))$ . Thus, by the induction hypothesis and Proposition 4.5, we have

$$ \begin{align*} \mathrm{reg}((I(G)^{[k]}+I_{S,B}): u_lv_{l,t})&\le \mathrm{aim}(\Gamma,k-1)+k-1\\ &\le \mathrm{aim}(G,k)+k-2. \end{align*} $$

Therefore, using inequality (15) and Lemma 2.1, we obtain

(16) $$ \begin{align} \mathrm{reg}((I(G)^{[k]}+I_{S,B})\le\mathrm{aim}(G,k)+k. \end{align} $$

This completes the proof of (ii). Now taking $S=\emptyset $ in inequality (16), we obtain $\mathrm {reg}((I(G)^{[k]})\le \mathrm {aim}(G,k)+k$ , and this proves (i).

Finally, the last statement follows immediately from (i) and Theorem 3.16. This completes the proof of the theorem.

5. Concluding remarks and some open problems

Let I be a (not necessarily square-free) monomial ideal and k a positive integer. Erey and Ficarra [Reference Erey and Ficarra12] recently defined the k-mathching power of I to be the ideals generated by $m_1m_2\cdots m_k$ where $m_1,m_2,\dots , m_k\in \mathcal {G}(I)$ have pairwise-disjoint supports. They denoted the kth matching power of I by $I^{[k]}$ as it is clear from the definition that if I is square-free, then the kth matching power of I is exactly its kth square-free power. Recall from [Reference Villarreal25, cf. Exercise 6.4.33] that $\mathrm {reg} I^{\mathcal {P}}=\mathrm {reg} I$ , where $I^{\mathcal {P}}$ , a square-free monomial ideal, denotes the polarization of I. By [Reference Erey and Ficarra12, Proposition 1.4], for any monomial ideal I and integer k, we have $\left (I^{[k]}\right )^{\mathcal {P}}=\left (I^{\mathcal {P}} \right )^{[k]}$ . Thus, one can use Theorem 3.11 to give a lower bound for the regularity of matching powers of monomial ideals. We leave the details to interested readers.

For several classes of graphs, the regularity of their edge ideals can be expressed in terms of the induced matching number of the corresponding graphs [Reference Banerjee, Beyarslan and Hà1]. For instance, $\mathrm {reg}(I(G))=\nu _1(G)+1$ if G belongs to one of the following classes:

(i) (weakly) chordal;
(ii) very well-covered;
(iii) sequentially Cohen–Macaulay bipartite;
(iv) $C_5$ -free vertex decomposable;
(v) certain unicyclic graphs.

For any graph G, it is well-known that $\mathrm {reg}(I(G)^k)\geq \nu _1(G)+2k-1$ . Moreover, it is expected that this lower bound will be attained for all the above-mentioned classes of graphs.

In the context of square-free powers, it was proved in Theorem 3.16 that $\mathrm {reg}(I(G)^{[k]})\geq \mathrm {aim} (G,k)+k$ for all $1\leq k\leq \nu (G)$ . Note that $\mathrm {aim}(G,1)=\nu _{1}(G)$ . Thus, $\mathrm {aim}(G,1)+1$ is indeed $\mathrm {reg}(I(G)^{[1]})$ when G is one of the above graphs. Hence, analogous to ordinary powers, we ask the following.

Question 5.1. Let G be a simple graph such that $\mathrm {reg}(I(G))=\nu _1(G)+1$ . Then is it true that $\mathrm {reg}(I(G)^{[k]})=\mathrm {aim}(G,k)+k$ for any $1\leq k\leq \nu (G)$ ?

We have an affirmative answer to the above question for block graphs, as seen in Proposition 4.11. Moreover, we have checked all chordal graphs up to eight vertices and observed that Proposition 5.1 has a positive answer. Based on all these, we conjecture the following.

Conjecture 5.2. Let G be a chordal graph. Then, for all $1\leq k\leq \nu (G)$ , we have

$$\begin{align*}\mathrm{reg}(I(G)^{[k]})=\mathrm{aim}(G,k)+k. \end{align*}$$

Acknowledgements

We are grateful to the referees for careful reading and their great feedback. We would like to thank the Infosys Foundation for a grant during our stay at the Chennai Mathematical Institute.

Funding information

K.S. also acknowledges support from the Anusandhan National Research Foundation (ANRF), Government of India, under the ARG-MATRICS Grant No. ANRF/ARGM/2025/002203/MTR.

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Figure 1 A block graph G.Figure 1 Long description.

Article contents

Admissible matchings and the Castelnuovo–Mumford regularity of square-free powers

Abstract

Keywords

MSC classification

Information

1. Introduction

2. Preliminaries

2.1. Combinatorics

2.2. Commutative algebra

Lemma 2.1 ([Reference Dao, Huneke and Schweig7, Lemma 2.10]).

Lemma 2.2 (Villarreal [Reference Villarreal25]).

3. k-admissible matching and a general lower bound

3.1. Existence of generalized k-admissible matching

Definition 3.4 ([Reference Francisco, Ha and Van Tuyl19, Definition 1.1]).

Definition 3.5 (cf. Eliahou and Kervaire [Reference Eliahou and Kervaire11]).

4. Square-free powers of block graphs

Proof of Claim 1.

Proof of Claim 2.

Proof of Claim 4.

5. Concluding remarks and some open problems

Acknowledgements

Funding information

References

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