$3$-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 $t\ge 8$.

21 pages

References in corpus (1)

• Four-coloring $P_6$-free graphs. II. Finding an excellent precoloring