Odd Hadwiger's conjecture for the complements of Kneser graphs

TL;DR

This work proves the odd Hadwiger conjecture for the complements of Kneser graphs $\bar{K}(n,k)$ with $n \ge 2k \ge 4$ by constructing explicit strongly $1$-shallow $K_t$-minors of order at least $\chi(\bar{K}(n,k))$. It presents three constructive schemes (Constructions 1–3) to realize such minors and leverages a partitioning lemma for the clique families $\mathcal{A}_i(n,k)$ to keep the bags simple (stars or singletons). The authors derive precise lower bounds on the odd Hadwiger number $h_o(\bar{K}(n,k))$, show the gap $h_o - \chi$ grows as $\Omega(1.5^{k})$ in the regime $7 \le 2k+1 \le n \le 3k-1$, and provide a detailed case analysis (covering $s=n/k$ values and $k$ small or large) to validate the conjecture across all admissible parameters. Overall, the paper advances understanding of odd Hadwiger questions for a large and structurally rich graph class and gives explicit, verifiable constructions that tie chromatic and minor-based obstructions together. The results have implications for coloring of signed graphs and broaden the classes for which odd Hadwiger can be guaranteed via constructive minors.

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 $χ(\overline{K}(n,k))$, 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 $Ω(1.5^{k})$.

Theorems & Definitions (23)

• Conjecture 1: Hadwiger's conjecture
• Conjecture 2: Odd Hadwiger's conjecture
• Definition 1
• Conjecture 3: Odd Hadwiger's conjecture restated
• Theorem 4
• Lemma 5
• proof
• Lemma 6
• proof
• Lemma 7