Dominating Hadwiger's Conjecture holds for all $2K_2$-free graphs

Abstract

A dominating $K_t$ minor in a graph $G$ is a sequence $(T_1,\dots,T_t)$ of pairwise disjoint non-empty connected subgraphs of $G$, such that for $1 \leq i<j\leq t$, every vertex in $T_j$ has a neighbor in $T_i$. Replacing ``every vertex in $T_j$'' by ``some vertex in $T_j$'' retrieves the standard definition of a $K_t$ minor. The strengthened notion was introduced by Illingworth and Wood [arXiv:2405.14299], who asked whether every graph with chromatic number $t$ contains a dominating $K_t$ minor. This is a substantial strengthening of the celebrated Hadwiger's Conjecture, which asserts that every graph with chromatic number $t$ contains a $K_t$ minor. At the ``New Perspectives in Colouring and Structure'' workshop held at the Banff International Research Station from September 29 - October 4, 2024, Norin referred to this question as the ``Dominating Hadwiger's Conjecture'' and believes it is likely false. In this paper we prove that the Dominating Hadwiger's Conjecture holds for all $2K_2$-free graphs. A key component of our proof is the clever use of the existence of an induced banner, obtained by adding a vertex adjacent to exactly one vertex on a cycle of length four.