Parameterized complexity of $k$-Coloring in graphs with no long induced paths

Abstract

We study the parameterized complexity of (List) $k$-Coloring in $H$-free graphs, where $H$ is a linear forest, that is, a disjoint union of paths. First, considering $k$ as the parameter, we establish the following: * For any $s\ge 0$, List $k$-Coloring in $(P_4+s{P}_1)$-free graphs is fixed-parameter tractable (FPT). * $k$-Coloring is W[1]-hard in $2P_2$-free graphs. The second result settles, in a strong form, a long-standing open problem posed by Hoàng, Kamiński, Lozin, Sawada, and Shu [Algorithmica, 2010]. Next, we prove that $k$-Coloring is NP-hard in $(P_4+P_2)$-free graphs. These three findings, together with known results from classical, non-parameterized complexity, yield a complete complexity classification of $k$-Coloring and List $k$-Coloring in $H$-free graphs, parameterized by $k$, into the cases: FPT, XP but W[1]-hard, and paraNP-hard. We also prove that $3$-Coloring is W[1]-hard in $P_t$-free graphs when parameterized by $t$. This answers a question of Golovach, Johnson, Paulusma, and Song [Journal of Graph Theory, 2017]. Finally, as a byproduct of our algorithm for List $k$-Coloring in $(P_4+s{P}_1)$-free graphs, we show that, for every fixed $s$ and $k$, there are only finitely many $(P_4+s{P}_1)$-free minimal obstructions to $k$-colorability. This settles a conjecture of Cameron, Hoàng, and Sawada [Disc. Appl. Math., 2022] and completes the dichotomy concerning the finiteness of the family of vertex-$k$-critical $H$-free graphs for every graph $H$ and every $k$.