On 3-colourability of $(bull,H)$-free graphs

The $3$-colourability problem is a well-known NP-complete problem and it remains NP-complete for $bull$-free graphs, where $bull$ is the graph consisting of $K_3$ with two pendant edges attached to two of its vertices. In this paper we study $3$-colourability of $(bull,H)$-free graphs for several graphs $H$. We show that these graphs are $3$-colourable or contain an induced odd wheel $W_{2p+1}$ for some $p\ge 2$ or a spindle graph $M_{3p+1}$ for some $p\ge 1$. Moreover, for all our results we can provide certifying algorithms that run in polynomial time.