# 3-Colouring P_t-free graphs without short odd cycles

For any odd t≥ 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≥ 8.

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