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Conflict-Free Colourings of Graphs and Hypergraphs

Published online by Cambridge University Press:  01 September 2009

JÁNOS PACH  and
GÁBOR TARDOS
JÁNOS PACH
Affiliation:
EPFL-SB-IMB-DCG, CH-1015 Lausanne, Switzerland and Department of Computer Science, City College, 138th Street at Convent Avenue, NY, NY 10031, USA (e-mail: pach@cims.nyu.edu)
GÁBOR TARDOS
Affiliation:
School of Computing Science, Simon Fraser University, 8888 University Drive, Burnaby, BC, V5A 1S6, Canada and Rényi Institute, 13–15 Reáltanoda utca Budapest, Hungary (e-mail: tardos@cs.sfu.edu)
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Abstract

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A colouring of the vertices of a hypergraph H is called conflict-free if each hyperedge E of H contains a vertex of ‘unique’ colour that does not get repeated in E. The smallest number of colours required for such a colouring is called the conflict-free chromatic number of H, and is denoted by χCF(H). This parameter was first introduced by Even, Lotker, Ron and Smorodinsky (FOCS 2002) in a geometric setting, in connection with frequency assignment problems for cellular networks. Here we analyse this notion for general hypergraphs. It is shown that $\chi_{\rm CF}(H)\leq 1/2+\sqrt{2m+1/4}$, for every hypergraph with m edges, and that this bound is tight. Better bounds of the order of m1/t log m are proved under the assumption that the size of every edge of H is at least 2t − 1, for some t ≥ 3. Using Lovász's Local Lemma, the same result holds for hypergraphs in which the size of every edge is at least 2t − 1 and every edge intersects at most m others. We give efficient polynomial-time algorithms to obtain such colourings.

Our machinery can also be applied to the hypergraphs induced by the neighbourhoods of the vertices of a graph. It turns out that in this case we need far fewer colours. For example, it is shown that the vertices of any graph G with maximum degree Δ can be coloured with log2+ε Δ colours, so that the neighbourhood of every vertex contains a point of ‘unique’ colour. We give an efficient deterministic algorithm to find such a colouring, based on a randomized algorithmic version of the Lovász Local Lemma, suggested by Beck, Molloy and Reed. To achieve this, we need to (1) correct a small error in the Molloy–Reed approach, (2) restate and re-prove their result in a deterministic form.

Information

Type
Paper
Information
Combinatorics, Probability and Computing , Volume 18 , Issue 5: New Directions in Algorithms, Combinatorics and Optimization , September 2009 , pp. 819 - 834
Copyright
Copyright © Cambridge University Press 2009

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