Maximum list $r$-colorable induced subgraphs in $kP_3$-free graphs

TL;DR

This paper presents a polynomial-time algorithm for the Max-Weight List r-Colorable Induced Subgraph problem on kP3-free graphs, generalizing several classical problems and advancing complexity classifications for related graph problems.

Contribution

It introduces a polynomial-time algorithm for a generalized coloring problem on kP3-free graphs, extending known results and providing new complexity insights.

Findings

Polynomial-time algorithm for Max-Weight List r-Colorable Induced Subgraph on kP3-free graphs.

Characterization of H-free graphs where the problem is polynomial-time solvable.

Progress towards a complexity dichotomy for Odd Cycle Transversal on H-free graphs.

Abstract

We show that, for every fixed positive integers $r$ and $k$, \textsc{Max-Weight List $r$-Colorable Induced Subgraph} admits a polynomial-time algorithm on $kP_3$-free graphs. This problem is a common generalization of \textsc{Max-Weight Independent Set}, \textsc{Odd Cycle Transversal} and \textsc{List $r$-Coloring}, among others. Our result has several consequences. First, it implies that, for every fixed $r \geq 5$, assuming $\mathsf{P}\neq \mathsf{NP}$, \textsc{Max-Weight List $r$-Colorable Induced Subgraph} is polynomial-time solvable on $H$-free graphs if and only if $H$ is an induced subgraph of either $kP_3$ or $P_5+kP_1$, for some $k \geq 1$. Second, it makes considerable progress toward a complexity dichotomy for \textsc{Odd Cycle Transversal} on $H$-free graphs, allowing to answer a question of Agrawal, Lima, Lokshtanov, Rz{\k{a}}{\.z}ewski, Saurabh, and Sharma [TALG 2024]. Third, it gives a short and self-contained proof of the known result of Chudnovsky, Hajebi, and Spirkl [Combinatorica 2024] that \textsc{List $r$-Coloring} on $kP_3$-free graphs is polynomial-time solvable for every fixed $r$ and $k$. We also consider two natural distance-$d$ generalizations of \textsc{Max-Weight Independent Set} and \textsc{List $r$-Coloring} and provide polynomial-time algorithms on $kP_3$-free graphs for every fixed integers $r$, $k$, and $d \geq 6$.