Graph Structure and Coloring

We denote by G=(V,E) a graph with vertex set V and edge set E. A graph G is claw-free if no vertex of G has three pairwise nonadjacent neighbours. Claw-free graphs are a natural generalization of line graphs. This thesis answers several questions about claw-free graphs and line graphs. In 1988, Chvatal and Sbihi proved a decomposition theorem for claw-free perfect graphs. They showed that claw-free perfect graphs either have a clique-cutset or come from two basic classes of graphs called elementary and peculiar graphs. In 1999, Maffray and Reed successfully described how elementary graphs can be built using line graphs of bipartite graphs and local augmentation. However gluing two claw-free perfect graphs on a clique does not necessarily produce claw-free graphs. The first result of this thesis is a complete structural description of claw-free perfect graphs. We also give a construction for all perfect circular interval graphs. This is joint work with Chudnovsky. Erdos and Lovasz conjectured in 1968 that for every graph G and all integers s,t≥ 2 such that s+t-1=χ(G) > ω(G), there exists a partition (S,T) of the vertex set of G such that ω(G|S)≥ s and χ(G|T)≥ t. This conjecture is known in the graph theory community as the Erdos-Lovasz Tihany Conjecture. For general graphs, the only settled cases of the conjecture are when s and t are small. Recently, the conjecture was proved for a few special classes of graphs: graphs with stability number 2, line graphs and quasi-line graphs. The second part of this thesis considers the conjecture for claw-free graphs and presents some progresses on it. This is joint work with Chudnovsky and Fradkin. Reed's ω, ∆, χ conjecture proposes that every graph satisfies χ≤ ⎡½ (Δ+1+ω)⎤ ; it is known to hold for all claw-free graphs. The third part of this thesis considers a local strengthening of this conjecture. We prove the local strengthening for line graphs, then note that previous results immediately tell us that the local strengthening holds for all quasi-line graphs. Our proofs lead to polytime algorithms for constructing colorings that achieve our bounds: The complexity are O(n²) for line graphs and O(n³m²) for quasi-line graphs. For line graphs, this is faster than the best known algorithm for constructing a coloring that achieves the bound of Reed's original conjecture. This is joint work with Chudnovsky, King and Seymour

Fractional Colouring and Hadwiger's Conjecture

AbstractLetGbe a loopless graph with noKp+1minor. We prove that the “fractional chromatic number” ofGis at most 2p; that is, it is possible to assign a rationalq(S)⩾0 to every stable setS⊆V(G) so that ∑S∋vq(S)=1 for every vertexv, and ∑Sq(S)⩽2p

Strong chromatic index and Hadwiger number

Abstract We investigate the effect of a fixed forbidden clique minor upon the strong chromatic index, both in multigraphs and in simple graphs. We conjecture for each that any ‐minor‐free multigraph of maximum degree has strong chromatic index at most . We present a construction certifying that if true the conjecture is asymptotically sharp as . In support of the conjecture, we show it in the case and prove the statement for strong clique number in place of strong chromatic index. By contrast, we make a basic observation that for ‐minor‐free simple graphs, the problem of strong edge‐colouring is “between” Hadwiger's Conjecture and its fractional relaxation. For , we also show that ‐minor‐free multigraphs of edge‐diameter at most 2 have strong clique number at most