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

2026-08-18

AI Summary

本文研究了无长诱导路径图中k-着色问题的参数化复杂性，通过证明两个W[1]-难问题，解决了关于该问题计算复杂性的长期开放问题。

Abstract

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 Hoàng, Kamiński, Lozin, Sawada, and Shu [Algorithmica, 2010]. The second result answers a question of Golovach, Johnson, Paulusma, and Song [Journal of Graph Theory, 2017].

Problem

• k-Coloring
• Parameterized Complexity
• H-free Graphs
• Induced Paths

Innovation

• parameterized complexity
• k-Coloring
• linear forest
• W[1]-hard
• Exponential Time Hypothesis (ETH)