Hadwiger's conjecture and its variants

Abstract

Hadwiger's conjecture, which is one of the infamous conjectures in graph theory, states that for $t \geq 1$, every graph with no $K_t$ as a minor is (t-1)-colourable. Gerards and Seymour strengthened this conjecture, that every graph with no $K_t$ as an odd minor is (t-1)-colourable. We are interested in variants of both conjectures, in terms of an improper colouring: for $t \geq 1$, is there an integer D such that every graph G with no $K_t$ minor (odd minor) has a vertex partition into k(t) parts so that every subgraph induced on each partition has the maximum degree at most D? For graphs with no $K_t$ minor, Edwards, Kim, Oum, Seymour, and the author proved k(t) = t-1 for $t \geq 1$, and this is sharp. With the essentially same proof, this holds for graphs with no bipartite $K_t$ subdivision. For graphs with no odd $K_t$ -minor, Oum and the author proved k(t) = 7t - 10 for $t \geq 2$. Using some previous results, this improves the result by Kawarabayashi.