Claw-free graphs and two conjectures on omega, Delta, and chi

This thesis concerns the relationship between four graph invariants: o, chi f, chi, and Delta. These are the clique number, the fractional chromatic number, the chromatic number, and the maximum degree, respectively 1. Trivially o ≤ chif ≤ chi ≤ Delta +1. We seek to improve the upper bound on chi. We are motivated by a conjecture of Reed, which essentially states that chi is at most the average of its trivial upper and lower bounds: Conjecture. For any graph, chi ≤ &ceill0; 1/2(Delta + 1 + o) &ceilr0; .;We begin by showing that much of the early evidence supporting the Main Conjecture also supports the Local Strengthening. In particular, the variant of the Local Strengthening obtained by replacing chi by chi f holds, as does the Local Strengthening when the stability number is two.;Guided by the first of these results we look towards line graphs, for which chif and chi agree asymptotically. We prove the Main Conjecture for line graphs, then we seek to generalize this result.;To do this we use recent results of Chudnovsky and Seymour, who characterized the structure of all claw-free graphs. We refine their results by introducing a graph reduction on certain types of homogeneous pairs of cliques that preserves the chromatic number. Thus we need only consider the problem of colouring skeletal claw-free graphs, which cannot be reduced. The structure of skeletal claw-free graphs is simpler than that of general claw-free graphs.;We call this the Main Conjecture, and propose a Local Strengthening based on the neighbourhood of a single vertex: Conjecture. For any graph G, chi ≤ maxv ∈V(G ) &ceill0; 1/2(d(v) + 1 + o( G[N¯(v)])) &ceilr0; .;We generalize two results from line graphs to the class of quasi-line graphs. Namely, that the Main Conjecture holds, and that chi f and chi agree asymptotically. We then consider all claw-free graphs. We prove the Main Conjecture for all claw-free graphs and we prove the Local Strengthening for claw-free graphs with a three-colourable complement. Our proofs yield polynomial-time colouring algorithms that achieve the conjectured bounds.;1We use standard graph theory notation, which can be found in the glossary.