Seymour and Woodall's conjecture holds for graphs with independence number two

TL;DR

This paper proves Seymour and Woodall's conjecture for graphs with independence number two by demonstrating the existence of specific minors related to the chromatic number.

Contribution

It establishes that for graphs with independence number two, a particular class of minors exists, confirming the conjecture in this case.

Findings

Seymour and Woodall's conjecture holds for graphs with independence number two.

Graphs with independence number two contain specific complete bipartite minors.

The paper introduces the graph $K^{ ext{ell}}_{ ext{ell}, ext{chi}(G)- ext{ell}}$ minors in this context.

Abstract

Woodall (and Seymour independently) in 2001 proposed a conjecture that every graph $G$ contains every complete bipartite graph on $\chi(G)$ vertices as a minor, where $\chi(G)$ is the chromatic number of $G$. In this paper, we prove that for each positive integer $\ell$ with $2\ell \leq \chi(G)$, each graph $G$ with independence number two contains a $K^{\ell}_{\ell,\chi(G)-\ell}$-minor, implying that Seymour and Woodall's conjecture holds for graphs with independence number two, where $K^{\ell}_{\ell,\chi(G)-\ell}$ is the graph obtained from $K_{\ell,\chi(G)-\ell}$ by making every pair of vertices on the side of the bipartition of size $\ell$ adjacent.