3-Colouring $$P_t$$-Free Graphs Without Short Odd Cycles

Abstract

For any odd $$t\ge 9$$ , we present a polynomial-time algorithm that solves the 3-colouring problem, and finds a 3-colouring if one exists, in $$P_{t}$$ -free graphs of odd girth at least $$t-2$$ . In particular, our algorithm works for $$(P_9, C_3, C_5)$$ -free graphs, thus making progress towards determining the complexity of 3-colouring in $$P_t$$ -free graphs, which is open for $$t\ge 8$$ .