Hadwiger's conjecture for graphs with forbidden holes

TL;DR

This paper provides new evidence supporting Hadwiger's conjecture by proving it holds for graphs with certain forbidden hole lengths related to their independence number, and also confirms the odd Hadwiger's conjecture under similar conditions.

Contribution

The paper proves Hadwiger's conjecture for graphs with no holes of specific lengths related to their independence number, and establishes the odd Hadwiger's conjecture for such graphs.

Findings

Hadwiger's conjecture holds for graphs with no holes of length between 4 and 2α(G)-1.

Graphs with no holes of length between 4 and 2α(G) contain an odd clique minor of size χ(G).

Provides evidence supporting Hadwiger's and odd Hadwiger's conjectures for restricted classes of graphs.

Abstract

Given a graph $G$, the Hadwiger number of $G$, denoted by $h(G)$, is the largest integer $k$ such that $G$ contains the complete graph $K_k$ as a minor. A hole in $G$ is an induced cycle of length at least four. Hadwiger's Conjecture from 1943 states that for every graph $G$, $h(G)\ge \chi(G)$, where $\chi(G)$ denotes the chromatic number of $G$. In this paper we establish more evidence for Hadwiger's conjecture by showing that if a graph $G$ with independence number $\alpha(G)\ge3$ has no hole of length between $4$ and $2\alpha(G)-1$, then $h(G)\ge\chi(G)$. We also prove that if a graph $G$ with independence number $\alpha(G)\ge2$ has no hole of length between $4$ and $2\alpha(G)$, then $G$ contains an odd clique minor of size $\chi(G)$, that is, such a graph $G$ satisfies the odd Hadwiger's conjecture.