Asymptotic Equivalence of Hadwiger's Conjecture and its Odd Minor-Variant
Abstract
Hadwiger's conjecture states that every Kt-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 Kt-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 Kt-minor is f(t)-colorable, then every graph without an odd Kt-minor is 2f(t)-colorable. Using this, the recent O(t t)-upper bound of Delcourt and Postle for the chromatic number of Kt-minor free graphs directly carries over to the chromatic number of odd Kt-minor-free graphs. This (slightly) improves a previous bound of O(t( t)2) for this problem by Delcourt and Postle.