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It is shown that, for every fixed $s$ and $k, there are only finitely many $(P_4+sP_1)$-free minimal obstructions to $k$-colorability.
We study the parameterized complexity of $k$-Coloring in $H$-free graphs, when $H$ is a linear forest (i.e., a disjoint union of paths) as an induced subgraph. We show two hardness results: * $k$-Coloring is W[1]-hard in $2P_2$-free graphs when parameterized by $k$. * $3$-Coloring is W[1]-hard in $P_t$-free graphs when parameterized by $t$. Moreover, assuming the ETH, these problems admit no algorithms solving $n$-vertex instances in time $f(k) \cdot n^{o(k)}$ and $f(t) \cdot n^{o(t/\log t)}$, respectively, for any computable function $f$. The first result resolves in a strong form a long-standing open problem, originally posed by Ho\`ang, Kami\'nski, Lozin, Sawada, and Shu [Algorithmica, 2010]. The second result answers a question of Golovach, Johnson, Paulusma, and Song [Journal of Graph Theory, 2017].
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