Odd Hadwiger's conjecture for the complements of Kneser graphs

TL;DR

This paper proves the odd Hadwiger's conjecture for the complements of Kneser graphs, extending previous results and providing new bounds on the odd Hadwiger number relative to the chromatic number.

Contribution

It establishes the validity of odd Hadwiger's conjecture for complements of Kneser graphs and introduces bounds on the odd Hadwiger number for these graphs.

Findings

Odd Hadwiger's conjecture holds for complements of Kneser graphs.

Contains a 1-shallow complete minor with size at least the chromatic number.

Shows exponential gap between odd Hadwiger number and chromatic number in certain cases.

Abstract

A generalization of the four-color theorem, Hadwiger's conjecture is considered as one of the most important and challenging problems in graph theory, and odd Hadwiger's conjecture is a strengthening of Hadwiger's conjecture by way of signed graphs. In this paper, we prove that odd Hadwiger's conjecture is true for the complements $\overline{K}(n,k)$ of the Kneser graphs $K(n,k)$, where $n\geq 2k \ge 4$. This improves a result of G. Xu and S. Zhou (2017) which states that Hadwiger's conjecture is true for this family of graphs. Moreover, we prove that $\overline{K}(n,k)$ contains a 1-shallow complete minor of a special type with order no less than the chromatic number $\chi(\overline{K}(n,k))$, and in the case when $7 \le 2k+1 \le n \le 3k-1$ the gap between the odd Hadwiger number and chromatic number of $\overline{K}(n,k)$ is $\Omega(1.5^{k})$.