Colouring Graphs of Bounded Diameter in the Absence of Small Cycles

Abstract: For $k\geq 1$, a $k$-colouring $c$ of $G$ is a mapping from $V(G)$ to ${1,2,\ldots,k}$ such that $c(u)\neq c(v)$ for any two non-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 ${\cal H}$, a graph $G$ is ${\cal H}$-free if $G$ does not contain any graph from ${\cal H}$ as an induced subgraph. Let $C_s$ 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 $(C_3,\ldots,C_s)$-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 $C_s$-free graphs of diameter $2$ and $(C_4,C_s)$-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 $4$.