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arXiv · 2008.04845

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

Alberto Rojas·Maya Stein
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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 $t\ge 8$.

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Alberto Rojas, Maya Stein. 2020-08-11. $3$-Colouring $P_t$-free graphs without short odd cycles. https://arxiv.org/abs/2008.04845

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