Chromatic Thresholds of Regular Graphs with Small Cliques

The chromatic threshold of a class of graphs is the value θ such that any graph in this class with a minimum degree greater than θn has a bounded chromatic number. Several important results related to the chromatic threshold of triangle-free graphs have been reached in the last 13 years, culminating in a result by Brandt and Thomassé stating that any triangle-free graph on n vertices with minimum degree exceeding 1/3 n has chromatic number at most 4. In this paper, the researcher examines the class of triangle-free graphs that are additionally regular. The researcher finds that any triangle-free graph on n vertices that is regular of degree (1/4+a)n with a \u3e 0 has chromatic number bounded by f (a), a function of a independent of the order of the graph n. After obtaining this result, the researcher generalizes this method to graphs that are free of larger cliques in order to limit the possible values of the chromatic threshold for regular Kr-free graphs

Vertex colouring and forbidden subgraphs - a survey

There is a great variety of colouring concepts and results in the literature. Here our focus is to survey results on vertex colourings of graphs defined in terms of forbidden induced subgraph conditions

Graph Theory

This is the report on an Oberwolfach conference on graph theory, held 16-22 January 2005. There were three main components to the event: 5-minute presentations, lectures, and workshops. All participants were asked to give a 5-minute presentation of their interests on the first day, and subsequent days were divided into lectures and workshops. The latter ranged over many different topics, but the main three topics were: infinite graphs, topological methods and their use to prove theorems in graph theory, and Rota’s conjecture for matroids

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

Colouring Cayley Graphs

We will discuss three ways to bound the chromatic number on a Cayley graph. 1. If the connection set contains information about a smaller graph, then these two graphs are related. Using this information, we will show that Cayley graphs cannot have chromatic number three. 2. We will prove a general statement that all vertex-transitive maximal triangle-free graphs on n vertices with valency greater than n/3 are 3-colourable. Since Cayley graphs are vertex-transitive, the bound of general graphs also applies to Cayley graphs. 3. Since Cayley graphs for abelian groups arise from vector spaces, we can view the connection set as a set of points in a projective geometry. We will give a characterization of all large complete caps, from which we derive that all maximal triangle-free cubelike graphs on 2n vertices and valency greater than 2n/4 are either bipartite or 4-colourable