Forcing clique immersions through chromatic number

Abstract

Building on recent work of Dvořák and Yepremyan, we show that every simple graph of minimum degree 7t + 7 contains K t as an immersion and that every graph with chromatic number at least 3.54t + 4 contains K t as an immersion. We also show that every graph on n vertices with no stable set of size three contains K 2⌊n/5⌋ as an immersion.Hence (i) holds for X 3 .By the same arguments, (i) and (ii) hold for X 1 . This proves the claim. ♦ Claims 5.3 and 5.5 complete the proof of Theorem 1.6.

Large Immersions in Graphs with Independence Number 3 and 4

“…Theorem 2 (Gauthier, Le and Wollan [13]). Every n-vertex graph G contains an immersion of a clique on It seems natural to ask whether the bound from Theorem 2 can be improved without necessarily improving Gauthier, Le and Wollan's underlying result that relates the chromatic number and the size of clique immersions, as Duchet and Meyniel did in the context of graph minors.…”

“…Every n-vertex graph G contains an immersion of a clique on It seems natural to ask whether the bound from Theorem 2 can be improved without necessarily improving Gauthier, Le and Wollan's underlying result that relates the chromatic number and the size of clique immersions, as Duchet and Meyniel did in the context of graph minors. The first attempt in this direction (actually earlier than [13]) has been carried out by Vergara [27]. She conjectured that every n-vertex graph with independence number 2 contains an immersion of K ⌈ n 2 ⌉ , and showed that this conjecture is equivalent to Conjecture 1 for graphs of independence number 2.…”

“…. The best currently known lower bound for Conjecture 1 is due to Gauthier, Le and Wollan [13] who showed that every graph G contains an immersion of a clique on χ(G)−4…”

Improper colourings inspired by Hadwiger's conjecture

“…The best upper bound, due to Gauthier et al . , says that every ‐immersion‐free graph is properly ‐colourable.…”

“…Often motivated by this question, structural and colouring properties of graphs excluding a fixed immersion have recently been widely studied [9, 10, 13, 15, 20, 43, 51]. The best upper bound, due to Gauthier et al [19], says that every K t -immersion-free graph is properly (3.54t + 3)-colourable.…”