Colouring Generalized Claw-Free Graphs and Graphs of Large Girth: Bounding the Diameter
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For a fixed integer, the k-Colouring problem is to decide if the vertices of a graph can be coloured with at most k colours for an integer k, such that no two adjacent vertices are coloured alike. A graph G is H-free if G does not contain H as an induced subgraph. It is known that for all k≥ 3, the k-Colouring problem is NP-complete for H-free graphs if H contains an induced claw or cycle. The case where H contains a cycle follows from the known result that the problem is NP-complete even for graphs of arbitrarily large fixed girth. We examine to what extent the situation may change if in addition the input graph has bounded diameter.
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