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

Abstract: For any odd

t g e 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 g e 8

.

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