An approximate version of Hadwiger's conjecture for claw-free
graphs

Hadwiger's conjecture states that if a graph is not
t-colorable then it contains the complete graph on t+1 vertices as a
minor.  The case t=4 is equivalent to the four color theorem and the
case t=5 was proved by Robertson, Seymour, and Thomas with the use of
the four color theorem.  For t>5, the conjecture remains open.  Reed and
Seymour have also proved that Hadwiger's conjecture holds for line
graphs and in a recent work with Maria Chudnovsky we proved that
Hadwiger's conjecture holds for a proper superset of the class of line
graphs, known as the class of quasi-line graphs (graphs where the
neighbor set of every vertex is the union of two cliques).

A graph is claw-free if it does not have a vertex with three pairwise
nonadjacent neighbors.  The class of claw-free graphs is a proper
superset of the class of quasi-line graphs.  In this talk I will outline
the proof of a weakened version of Hadwiger's conjecture for claw-free
graphs.  The proof uses a structure theorem from a recent work of
Chudnovsky and Seymour.  At the end of the talk I will discuss the
progress that has been made on the actual Hadwiger's conjecture for
claw-free graphs.

This is a joint work with Maria Chudnovsky.