Verifying Hadwiger's Conjecture for Examples of Graphs with $\alpha(G) = 2$

Abstract

Hadwiger's Conjecture states that every graph with chromatic number $k$ contains a complete graph on $k$ vertices as a minor. This conjecture is a tremendous strengthening of the Four-Colour Theorem and is regarded as one of the most important open problems in graph theory. The case of Hadwiger's Conjecture for graphs with $\alpha(G) = 2$ has garnered much attention. Seymour writes: ``My own belief is, if Hadwiger's Conjecture is true for graphs with stability number two then it is probably true in general, so it would be very nice to decide this case.'' This paper presents several tools useful for proving that a graph $G$ with $\alpha(G) = 2$ satisfies Hadwiger's Conjecture. In doing so, we survey and generalise several classical results on the $\alpha(G) = 2$ case of Hadwiger's Conjecture. Further, we apply these tools to prove variants of Hadwiger's Conjecture for several noteworthy classes of graphs with $\alpha(G) = 2$. In particular, we prove Hadwiger's Conjecture for inflations of the complements of the following graphs: graphs with girth at least $5$, triangle-free Kneser graphs, and the Clebsch, Mesner, and Gewirtz graphs. This paper also highlights classes of graphs with $\alpha(G) = 2$ where it is unknown if Hadwiger's Conjecture holds.