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

TL;DR

This paper investigates the 3-colourability of graphs free of certain subgraphs, establishing conditions under which such graphs are 3-colourable or contain specific induced subgraphs, with polynomial-time certifying algorithms.

Contribution

It characterizes 3-colourability in (bull, H)-free graphs and provides polynomial-time certifying algorithms for these cases.

Findings

Graphs are 3-colourable or contain an odd wheel or spindle graph.

Results apply to several graphs H, extending known NP-completeness results.

Algorithms for certifying 3-colourability are polynomial-time.

Abstract

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 $p\geq 2$ or a spindle graph $M_{3p+1}$ for some $p\geq 1$. Moreover, for all our results we can provide certifying algorithms that run in polynomial time.