Colouring Graphs of Bounded Diameter in the Absence of Small Cycles

For \(k\ge 1\), a k-colouring c of G is mapping from V(G) to \(\{1,2,\ldots ,k\}\) such that \(c(u)\ne c(v)\) for any two non-adjacent vertices u and v. The k -Colouring problem decide if graph has k-colouring. family graphs \(\mathcal{H}\), \(\mathcal{H}\)-free does not contain \(\mathcal{H}\) as an induced subgraph. Let \(C_s\) be the s-vertex cycle. In previous work (MFCS 2019) we examined effect bounding diameter on complexity 3-Colouring \((C_3,\ldots ,C_s)\)-free H-free where H some polyad. Here, prove certain small values s polynomial-time solvable \(C_s\)-free 2 \((C_4,C_s)\)-free 2. fact, our results hold more general List 3-Colouring. We complement these with hardness result 4.