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Maximum Partial List H-Coloring on P_5-free graphs in polynomial time

Daniel Lokshtanov, Paweł Rzążewski, Saket Saurabh, Roohani Sharma, Meirav Zehavi

TL;DR

It is shown that Maximum Partial List H-Coloring is polynomial-time solvable on P_5-free graphs for every fixed graph H, which implies that Maximum k-Colorable Subgraph is polynomial-time solvable on P_5-free graphs.

Abstract

In this article we show that Maximum Partial List H-Coloring is polynomial-time solvable on P_5-free graphs for every fixed graph H. In particular, this implies that Maximum k-Colorable Subgraph is polynomial-time solvable on P_5-free graphs. This answers an open question from Agrawal, Lima, Lokshtanov, Saurabh & Sharma [SODA 2024]. This also improves the $n^{ω(G)}$-time algorithm for Maximum Partial H-Coloring by Chudnovsky, King, Pilipczuk, Rzążewski & Spirkl [SIDMA 2021] to polynomial-time algorithm.

Maximum Partial List H-Coloring on P_5-free graphs in polynomial time

TL;DR

It is shown that Maximum Partial List H-Coloring is polynomial-time solvable on P_5-free graphs for every fixed graph H, which implies that Maximum k-Colorable Subgraph is polynomial-time solvable on P_5-free graphs.

Abstract

In this article we show that Maximum Partial List H-Coloring is polynomial-time solvable on P_5-free graphs for every fixed graph H. In particular, this implies that Maximum k-Colorable Subgraph is polynomial-time solvable on P_5-free graphs. This answers an open question from Agrawal, Lima, Lokshtanov, Saurabh & Sharma [SODA 2024]. This also improves the -time algorithm for Maximum Partial H-Coloring by Chudnovsky, King, Pilipczuk, Rzążewski & Spirkl [SIDMA 2021] to polynomial-time algorithm.

Paper Structure

This paper contains 5 sections, 9 theorems, 1 algorithm.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Solving the problem when there exists a connected solution
  4. Constructing a polynomial-sized family that contains subsets of connected components of a solution
  5. Proof of Theorem \ref{['thm:main']} by reduction to the Maximum Weight Independent Set problem on the blob graph

Key Result

Theorem 1

Maximum Partial List $H$-Coloring can be solved in $k^{\mathcal{O}(k)} \cdot n^{\mathcal{O}(k^3)}$ time on $P_5$-free graphs, where $k:=|V(H)|$.

Theorems & Definitions (17)

  • Theorem 1
  • Proposition 2.1: Theorem $8$, bacso1990dominating
  • Proposition 2.2: DBLP:conf/soda/LokshantovVV14, Independent Set on $P_5$-free
  • Lemma 1
  • proof
  • Claim 3.1
  • Lemma 2
  • Lemma 3
  • proof
  • Claim 4.1
  • ...and 7 more