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

Hadwiger's conjecture states that every Kt-minor free graph is (t?1)-colorable. A qualitative strengthening of this raised by Gerards and Seymour, known as the Odd conjecture, similarly with no odd For both conjectures, their asymptotic relaxations remain open, i.e., whether an upper bound on chromatic number form Ct for some constant C>0 exists. We show if without a f(t)-colorable, then 2f(t)-colorable. As direct corollary, we present new state art O(tlog?log?t) graphs Kt-minor. Moreover, short proof our result substantially simplifies proofs all previous bounds established in sequence papers during last 12 years.