6 research outputs found
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Edge-colourings of graphs
All the results in this thesis are concerned with the classification of graphs by their chromatic class.
We first extend earlier results of Fiorini and others to give a complete list of critical graphs of order at most ten. We give conditions for extending the edge-colouring of a nearly complete subgraph to the whole graph and use this result to prove a special case of Vizing's conjecture. We also use other methods to solve further cases of this conjecture.
A major part of the thesis classifies those graphs with at most 4 vertices of maximum degree and this work is generalised to graphs with r vertices of maximum degree. We also completely classify all regular graphs G with degree at least 6/7|V(G)|.
Finally we give some examples of even order critical graphs and introduce the concept of a supersnark
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The chromatic index of simple graphs
The object of this thesis is twofold:
(i) to study the structural properties of graphs which are critical with respect to edge-colourings;
(ii) to apply the results obtained to the classification problem arising from Vizing's Theorem.
Chapter 1 contains a historical, non-technical introduction, general graph-theoretic definitions and notation, a discussion of Vizing's Theorem as well as a survey of the main results obtained to date in Vizing's classification problem. Chapter 2 introduces the notion of criticality in the first section; the second section contains both well-known and new constructions of critical graphs which will be used in later chapters. The third and final section contains new results concerning elementary properties of critical graphs. Chapter 3 deals with uniquely-colourable graphs and their relationship to critical graphs. Chapter 4 contains results on the connectivity of critical graphs, whereas Chapter 5 deals with bounds on the number of edges of these graphs. In particular, bounds improving those given by Vizing are presented. These results are applied to problems concerning planar graphs. In Chapter 6, critical graphs of small order are discussed. All such graphs of order at most 8 are determined, while the 'critical graph conjecture’ of Beineke & Wilson and Jakobsen is shown to be true for all graphs on at most 10 vertices. The seventh and final chapter deals with circuit length properties of critical graphs. In particular, the minimal order of certain critical graphs with given girth and maximum valency is determined. Results improving Vizing’s estimate of the circumference of critical graphs are also given. The Appendix includes a computer programme which generates critical graphs from simpler ones using a constructive algorithm given in Chapter 2
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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