Dominating Hadwiger's Conjecture for graphs $G$ with $\alpha(G)=2$

Abstract

Hadwiger's Conjecture from 1943 states that every graph with chromatic number $t$ contains a $K_t$ minor. Illingworth and Wood [arXiv:2405.14299] introduced the concept of a ``dominating $K_t$ minor'' and asked whether every graph with chromatic number $t$ contains a dominating $K_t$ minor. This question is a substantial strengthening of Hadwiger's Conjecture. Norin referred to it as the ``Dominating Hadwiger's Conjecture'' and believes it is likely false. In this paper we first observe that a $t$-chromatic $G$ on $n$ vertices with independence number $\alpha(G)\le2$ contains a dominating $K_t$ minor if and only if $G$ contains a dominating $K_{\lceil n/2\rceil}$ minor. Building on this and using a deep result of Chudnovsky and Seymour on packing seagulls, we prove that every graph $G$ on $n$ vertices with $\alpha(G)\le 2$ and $2\omega(G)\ge \lceil n/2\rceil+1$ satisfies the Dominating Hadwiger's Conjecture, where $\omega(G)$ denotes the clique number of $G$. We further prove that every $H$-free graph $G$ with $\alpha(G)\le 2$ satisfies the Dominating Hadwiger's Conjecture, where $H\in\{2K_1+P_4, K_2+2K_2, K_2+(K_1\cup K_3), K_1+(K_1\cup K_5), W_5^<, W_5^-, W_5, K_7^<, K_7^-, K_7\}$, or $H\ne K_2\cup K_3$ is any graph on at most five vertices such that $\alpha(H)\le2$.