Asymptotic equivalence of Hadwiger's conjecture and its odd minor-variant
Raphael Steiner · Journal of Combinatorial Theory Series B · 2022
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. As a direct corollary, we present a new state of the art bound O(tloglogt) for the chromatic number of graphs with no odd Kt-minor. Moreover, the short proof of our result substantially simplifies the proofs of all the previous asymptotic bounds for the chromatic number of odd Kt-minor free graphs established in a sequence of papers during the last 12 years.