Vertex-critical $(P_5,\text{chair})$-free and $(P_5,\text{cricket})$-free graphs

For graphs $G, F_1$ and $F_2$, we say that $G$ is $(F_1,F_2)$-free if neither $F_1$ nor $F_2$ is an induced subgraph of $G$. We say that $G$ is $k$-vertex-critical if the chromatic number of $G$ is $k$, but every proper induced subgraph of $G$ has chromatic number at most $k-1$. The $\textit{chair}$ graph is a $5$-vertex graph obtained by adding a pendant vertex to one of the two central vertices of a path on $4$ vertices. The $\textit{cricket}$ graph is a $5$-vertex graph obtained by adding two pendant vertices to a common vertex of a triangle. The path on $5$ vertices is denoted by $P_5$. We prove that for every $k \geq 1$, there are only finitely many $(P_5,\text{chair})$-free $k$-vertex-critical graphs. We also prove that the same conclusion holds if $\text{chair}$ is replaced by $\text{cricket}$. We further characterize all $5$-vertex-critical $(P_5,\text{chair})$-free graphs, all $5$-vertex-critical $(P_5,\text{cricket})$-free graphs and all $6$-vertex-critical $(P_5,\text{cricket})$-free graphs. Our proofs rely on bounding the size of antichains and developing Ramsey-theoretic ideas. For any fixed integer $k \geq 1$, our results imply the existence of a polynomial time algorithm to decide whether a $(P_5,\text{chair})$-free (or $(P_5,\text{cricket})$-free) graph is $(k-1)$-colourable such that this algorithm can also present a negative constant-size certificate in case the graph is not $(k-1)$-colourable.