Asymptotic equivalence of Hadwiger's conjecture and its odd minor-variant

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. As a direct corollary, we present a new state of the art bound O(t log log t) for the chromatic number of graphs with no odd K t -minor. Moreover, the short proof of our result substantially simplifies the proofs of all the previous asymptotic bounds for the chromatic number of odd K t -minor free graphs established in a sequence of papers during the last 12 years.