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

TL;DR

This work investigates dominating Hadwiger-type questions for graphs with $\alpha(G)=2$, showing that such graphs satisfy a half-order dominating minor bound under natural parameters: specifically, if $2\omega(G) \ge \lceil n/2\rceil+1$, then $h_d(G) \ge \lceil n/2\rceil$, with the proof leveraging the deep seagull packing result of Chudnovsky and Seymour. It then extends these findings to broad $H$-free classes by proving that for $H$ in a list of small graphs (and $\alpha(H) \le 2$), the dominating Hadwiger bound holds, i.e., $h_d(G) \ge \lceil n/2\rceil$, and in turn yields $h_d(G) \ge \chi(G)$ in many cases. The approach combines structural lemmas for graphs with large minimum degree, the seagull framework, and careful case analyses, providing substantial progress toward the Dominating Hadwiger Conjecture in the $\alpha(G)=2$ regime and connecting to Ramsey-type constraints. These results advance understanding of dominating clique minors and offer new avenues for verifying Hadwiger-type conjectures in constrained graph classes.

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 $α(G)\le2$ 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 $α(G)\le 2$ and $2ω(G)\ge \lceil n/2\rceil+1$ satisfies the Dominating Hadwiger's Conjecture, where $ω(G)$ denotes the clique number of $G$. We further prove that every $H$-free graph $G$ with $α(G)\le 2$ 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 $α(H)\le2$.

Figures (2)

• Figure 1: Hammer and Kite
• Figure 2: Three graphs on eight vertices

Theorems & Definitions (43)

• Conjecture 1.1: Hadwiger's Conjecture Had43
• Conjecture 1.2: Dominating Hadwiger's Conjecture IW24
• Theorem 1.3: Song and Tibbetts SongTibbetts25
• Conjecture 1.4
• Theorem 1.5
• proof
• Theorem 1.6
• Corollary 1.7
• proof
• Corollary 1.8