100 research outputs found
Goldberg's Conjecture is true for random multigraphs
In the 70s, Goldberg, and independently Seymour, conjectured that for any
multigraph , the chromatic index satisfies , where . We show that their conjecture (in a
stronger form) is true for random multigraphs. Let be the probability
space consisting of all loopless multigraphs with vertices and edges,
in which pairs from are chosen independently at random with
repetitions. Our result states that, for a given ,
typically satisfies . In
particular, we show that if is even and , then
for a typical . Furthermore, for a fixed
, if is odd, then a typical has
for , and
for .Comment: 26 page
Extension from precoloured sets of edges
We consider precolouring extension problems for proper edge-colourings of graphs and multigraphs, in an attempt to prove stronger versions of Vizing's and Shannon's bounds on the chromatic index of (multi)graphs in terms of their maximum degree Ī. We are especially interested in the following question: when is it possible to extend a precoloured matching to a colouring of all edges of a (multi)graph? This question turns out to be related to the notorious List Colouring Conjecture and other classic notions of choosability
Graph Theory
Highlights of this workshop on structural graph theory included new developments on graph and matroid minors, continuous structures arising as limits of finite graphs, and new approaches to higher graph connectivity via tree structures
An extensive English language bibliography on graph theory and its applications, supplement 1
Graph theory and its applications - bibliography, supplement
Multigraphs with High Chromatic Index
In this thesis we take a specialized approach to edge-colouring by focusing exclusively on multigraphs with high chromatic index. The bulk of our results can be classified into three categories. First, we prove results which aim to characterize those multigraphs achieving known upper bounds. For example, Goldberg's Theorem says that Ļ'ā¤ Ī+1+(Ī-2}/(gā+1) (where Ļ' denotes chromatic index, Ī denotes maximum degree, and gā denotes odd girth). We characterize this bound by proving that for a connected multigraph G, Ļ'= Ī+1+(Ī-2}/(gā+1) if and only if G=Ī¼C_gā and (gā+1)|2(Ī¼-1) (where Ī¼ denotes maximum edge-multiplicity).
Our second category of results are new upper bounds for chromatic index in multigraphs, and accompanying polynomial-time edge-colouring algorithms. Our bounds are all approximations to the famous Seymour-Goldberg Conjecture, which asserts that Ļ'ā¤ max{āĻā, Ī+1} (where Ļ=max{(2|E[S]|)/(|S|-1): SāV, |S|ā„3 and odd}). For example, we refine Goldberg's classical Theorem by proving that Ļ'ā¤ max{āĻā, Ī+1+(Ī-3)/(gā+3)}.
Our third category of results are characterizations of high chromatic index in general, with particular focus on our approximation results. For example, we completely characterize those multigraphs with Ļ'> Ī+1+(Ī-3)/(gā+3).
The primary method we use to prove results in this thesis is the method of Tashkinov trees. We first solidify the theory behind this method, and then provide general edge-colouring results depending on Tashkinov trees. We also explore the limits of this method, including the possibility of vertex-colouring graphs which are not line graphs of multigraphs, and the importance of Tashkinov trees with regard to the Seymour-Goldberg Conjecture
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