Ks,t minors in (s + t)-chromatic graphs

Alexandr Kostochka In search of ways to attack Hadwiger’s Conjecture, Woodall and independently Seymour suggested to prove the weaker conjecture that Every (s + t)-chromatic graph has a Ks,t-minor. If the conjecture were true for all values of s and t, it would imply that for k ≥ 2 every (2k − 2)-chromatic graph has a Kk-minor. The conjecture is evident for s = 1. The validity of the conjecture for s = 2 and all t was proved by Woodall, and also follows from a result by Chudnovsky, Reed, and Seymour. Prince and the speaker proved the conjecture for s = 3 and t ≥ 6500. The aim of the talk is to show that for every fixed s and large t, the conjecture holds in a slightly stronger form: Let s and t be positive integers such that t > t0(s) := (240s log2 s) 8s log2 . Then every (s+t)-chromatic graph has a K∗ s,t-minor, where K ∗ s,t is obtained from Ks,t by adding all edges between the vertices of the partite set of size s. The result is sharp in the sense that for every s, t ≥ 3, there are infinitely many (s + t)critical graphs that do not have Ks,t+1-minors.