Colouring Cayley Graphs
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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 <i>n</i> vertices with valency greater than <i>n</i>/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 2<sup>n</sup> vertices and valency greater than 2<sup>n</sup>/4 are either bipartite or 4-colourable.
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Lei Chu (2005). Colouring Cayley Graphs. UWSpace. http://hdl.handle.net/10012/1125