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

TL;DR

The paper proves that Seymour and Woodall's conjecture holds for graphs with independence number at most $2$ by showing that for every $\ell$ with $2\ell \le \chi(G)$, such a graph $G$ satisfies $G \succeq_m K^{\ell}_{\ell,\chi(G)-\ell}$. The approach introduces a framework including capacity cap$(C)$, auxiliary vertex-set relations, and the five-wheel, combined with a minimal counterexample argument and a vertex-critical decomposition to construct the required minor via contracting disjoint $P_3$ paths and an edge. It also establishes the equivalence of related conjectures for this graph class. The results provide a concrete minor-construction method that advances Hadwiger-type questions for graphs constrained by a small independence number, reinforcing connections between chromatic number and minor containment in this setting.

Abstract

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

Theorems & Definitions (21)

• Conjecture 1.1
• Conjecture 1.2
• Conjecture 1.3
• Conjecture 1.4
• Conjecture 1.5
• Theorem 1.6
• Theorem 2.1
• Theorem 2.2
• Theorem 2.3
• Lemma 2.4