Hadwiger's Conjecture for inflations of 3-chromatic graphs

Hadwiger's Conjecture states that every k-chromatic graph has a complete minor of order k. A graph G′ is an inflation of a graph G if G′ is obtained from G by replacing each vertex v of G by a clique Cv and joining two vertices of distinct cliques by an edge if and only if the corresponding vertices of G are adjacent. We present an algorithm for computing an upper bound on the chromatic number χ(G′) of any inflation G′ of any 3-chromatic graph G. As a consequence, we deduce that Hadwiger's Conjecture holds for any inflation of any 3-colorable graph.