Hadwiger’s Conjecture and inflations of the Petersen graph

An inflation of a graph G is obtained by replacing vertices in G by disjoint cliques and adding all possible edges between any pair of cliques corresponding to adjacent vertices in G . We prove that the chromatic number of an arbitrary inflation F of the Petersen graph is equal to the chromatic number of some inflated 5-cycle contained in F . As a corollary, we find that Hadwiger’s Conjecture holds for any inflation of the Petersen graph. This solves a problem posed by Bjarne Toft.