Colouring graphs of bounded diameter in the absence of small cycles

Abstract

For k≥1, a k-colouring c of G is a mapping from V(G) to {1,2,…,k} such that c(u)≠c(v) for any two adjacent vertices u and v. The k-Colouring problem is to decide if a graph G has a k-colouring. For a family of graphs H, a graph G is H-free if G does not contain any graph from H as an induced subgraph. Let Cs be the s-vertex cycle. In previous work (MFCS 2019) we examined the effect of bounding the diameter on the complexity of 3-Colouring for (C3,…,Cs)-free graphs and H-free graphs where H is some polyad. Here, we prove for certain small values of s that 3-Colouring is polynomial-time solvable for Cs-free graphs of diameter 2 and (C4,Cs)-free graphs of diameter 2. In fact, our results hold for the more general problem List 3-Colouring. We complement these results with some hardness result for diameter 4.