Asymptotic Equivalence of Hadwiger's Conjecture and its Odd Minor-Variant

TL;DR

This paper proves that the asymptotic chromatic bounds for graphs excluding minors extend similarly to graphs excluding odd minors, linking Hadwiger's conjecture and its odd variant with improved bounds.

Contribution

It establishes that bounds on chromatic number for minor-free graphs imply similar bounds for odd minor-free graphs, connecting two major conjectures.

Findings

Asymptotic equivalence between Hadwiger's conjecture and its odd variant.

Extension of Delcourt and Postle's $O(t\,\log\log t)$ bound to odd minors.

Improvement over previous $O(t(\log \log t)^2)$ bound.

Abstract

Hadwiger's conjecture states that every $K_t$-minor free graph is $(t-1)$-colorable. A qualitative strengthening of this conjecture raised by Gerards and Seymour, known as the Odd Hadwiger's conjecture, states similarly that every graph with no odd $K_t$-minor is $(t-1)$-colorable. For both conjectures, their asymptotic relaxations remain open, i.e., whether an upper bound on the chromatic number of the form $Ct$ for some constant $C>0$ exists. We show that if every graph without a $K_t$-minor is $f(t)$-colorable, then every graph without an odd $K_t$-minor is $2f(t)$-colorable. Using this, the recent $O(t\log\log t)$-upper bound of Delcourt and Postle for the chromatic number of $K_t$-minor free graphs directly carries over to the chromatic number of odd $K_t$-minor-free graphs. This (slightly) improves a previous bound of $O(t(\log \log t)^2)$ for this problem by Delcourt and Postle.