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Clique Minors In Graphs And Their Complements (2000)
| Citations: | 2 - 0 self |
BibTeX
@MISC{Reed00cliqueminors,
author = {Bruce Reed and Robin Thomas},
title = {Clique Minors In Graphs And Their Complements },
year = {2000}
}
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Abstract
A graph H is a minor of a graph G if H can be obtained from a subgraph of G by contracting edges. Let t ≥ 1 be an integer, and let G be a graph on n vertices with no minor isomorphic to Kt+1. Kostochka conjectures that there exists a constant c = c(k) independent of G such that the complement of G has a minor isomorphic to Ks, wheres=⌈1 2 (1 + 1/t)n − c⌉. We prove that Kostochka’s conjecture is equivalent to the conjecture of Duchet and Meyniel that every graph with no minor isomorphic to Kt+1 has an independent set of size at least n/t. We deduce that Kostochka’s conjecture holds for all integers t ≤ 5, and that a weaker form with s replaced by s ′ = ⌈ 1 2 (1 + 1/(2t))n − c⌉ holds for all integers t ≥ 1.
