A note on Hadwiger's Conjecture for W5-free graphs with independence number two
Abstract
The Hadwiger number of a graph G, denoted h(G), is the largest integer t such that G contains Kt as a minor. A famous conjecture due to Hadwiger in 1943 states that for every graph G, h(G) (G), where (G) denotes the chromatic number of G. Let α(G) denote the independence number of G. A graph is H-free if it does not contain the graph H as an induced subgraph. In 2003, Plummer, Stiebitz and Toft proved that h(G) (G) for all H-free graphs G with α(G) 2, where H is any graph on four vertices with α(H) 2, H=C5, or H is a particular graph on seven vertices. In 2010, Kriesell considered a particular strengthening of Hadwiger's conjecture due to Seymour and subsequently generalized the statement to include all forbidden subgraphs H on five vertices with α(H) 2. In this note, we prove that h(G) (G) for all W5-free graphs G with α(G) 2, where W5 denotes the wheel on six vertices.