|Graphs of low average degree without independent transversals
|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
|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.
|Appears in Collections:
|Články / Articles (KMA)
Please use this identifier to cite or link to this item:
Items in DSpace are protected by copyright, with all rights reserved, unless otherwise indicated.