Title: Graphs of low average degree without independent transversals
Authors: Groenland, Carla
Kaiser, Tomáš
Treffers, Oscar
Wales, Matthew
Citation: GROENLAND, C. KAISER, T. TREFFERS, O. WALES, M. Graphs of low average degree without independent transversals. Journal of Graph Theory, 2023, roč. 102, č. 2, s. 374-387. ISSN: 0364-9024
Issue Date: 2023
Publisher: Wiley
Document type: článek
URI: 2-s2.0-85135964681
ISSN: 0364-9024
Keywords in different language: independent transversal;Lovász Local Lemma;entropy compression
Abstract in different language: An independent transversal of a graph G with a vertex partition P is an independent set of G intersecting each block of P in a single vertex. Wanless and Wood proved that if each block of P has size at least t and the average degree of vertices in each block is at most t/4, then an independent transversal of P exists. We present a construction showing that this result is optimal: for any ε>0 and sufficiently large t, there is a family of forests with vertex partitions whose block size is at least t, average degree of vertices in each block is at most (1/4+ε)t, and there is no independent transversal. This unexpectedly shows that methods related to entropy compression such as the Rosenfeld-Wanless-Wood scheme or the Local Cut Lemma are tight for this problem. Further constructions are given for variants of the problem, including the hypergraph version.
Rights: © authors
Appears in Collections:Články / Articles (KMA)

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