A note on Hadwiger's Conjecture for $W_5$-free graphs with independence number two

TL;DR

This paper proves that Hadwiger's conjecture holds for all $W_5$-free graphs with independence number two, extending known results for certain classes of graphs and contributing to the understanding of graph minors and chromatic number relations.

Contribution

It establishes the validity of Hadwiger's conjecture for $W_5$-free graphs with independence number two, a new class not previously covered.

Findings

Proves $h(G) geq \, ext{chromatic number}$ for $W_5$-free graphs with $ ext{independence number} \, 2$

Extends the class of graphs for which Hadwiger's conjecture is verified

Builds on prior work for graphs with independence number at most 2

Abstract

The Hadwiger number of a graph $G$, denoted $h(G)$, is the largest integer $t$ such that $G$ contains $K_t$ as a minor. A famous conjecture due to Hadwiger in 1943 states that for every graph $G$, $h(G) \ge \chi(G)$, where $\chi(G)$ denotes the chromatic number of $G$. Let $\alpha(G)$ denote the independence number of $G$. A graph is $H$-free if it does not contain the graph $H$ as an induced subgraph. In 2003, Plummer, Stiebitz and Toft proved that $h(G) \ge \chi(G)$ for all $H$-free graphs $G$ with $\alpha(G) \le 2$, where $H$ is any graph on four vertices with $\alpha(H) \le 2$, $H=C_5$, or $H$ is a particular graph on seven vertices. In 2010, Kriesell considered a particular strengthening of Hadwiger's conjecture due to Seymour and subsequently generalized the statement to include all forbidden subgraphs $H$ on five vertices with $\alpha(H) \le 2$. In this note, we prove that $h(G) \ge \chi(G)$ for all $W_5$-free graphs $G$ with $\alpha(G) \le 2$, where $W_5$ denotes the wheel on six vertices.