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Detachments of Hypergraphs I: The Berge–Johnson Problem

Published online by Cambridge University Press:  27 February 2012

M. A. BAHMANIAN
M. A. BAHMANIAN*
Affiliation:
Department of Mathematics and Statistics, Auburn University, Auburn, AL 36849-5310, USA (e-mail: mzb0004@auburn.edu)
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Abstract

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A detachment of a hypergraph is formed by splitting each vertex into one or more subvertices, and sharing the incident edges arbitrarily among the subvertices. For a given edge-coloured hypergraph , we prove that there exists a detachment such that the degree of each vertex and the multiplicity of each edge in (and each colour class of ) are shared fairly among the subvertices in (and each colour class of , respectively).

Let $(\lambda_1,\ldots,\lambda_m) K^{h_1,\ldots,\, h_m}_{p_1,\ldots,\, p_n}$ be a hypergraph with vertex partition {V1,. . .,Vn}, |Vi| = pi for 1 ≤ i ≤ n such that there are λi edges of size hi incident with every hi vertices, at most one vertex from each part for 1 ≤ i ≤ m (so no edge is incident with more than one vertex of a part). We use our detachment theorem to show that the obvious necessary conditions for $(\lambda_1\dots,\lambda_m) K^{h_1,\ldots,h_m}_{p_1,\ldots,p_n}$ to be expressed as the union 1 ∪ ··· ∪ k of k edge-disjoint factors, where for 1 ≤ i ≤ k, i is ri-regular, are also sufficient. Baranyai solved the case of h1 = ··· = hm, λ1 = ··· = λm = 1, p1 = ··· = pm, r1 = ··· = rk. Berge and Johnson (and later Brouwer and Tijdeman, respectively) considered (and solved, respectively) the case of hi = i, 1 ≤ i ≤ m, p1 = ··· = pm = λ1 = ··· = λm = r1 = ··· = rk = 1. We also extend our result to the case where each i is almost regular.

Keywords

05C6505C1505C7005C5105B4005B05

Information

Type
Paper
Information
Combinatorics, Probability and Computing , Volume 21 , Issue 4 , July 2012 , pp. 483 - 495
Copyright
Copyright © Cambridge University Press 2012

References

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