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Coloring Hardness on Low Twin-Width Graphs

Édouard Bonnet

TL;DR

This work investigates coloring problems on graphs of bounded twin-width, establishing a sharp hardness boundary: Min Coloring is NP-hard on $\mathcal{T}_3$ and, for every fixed $k \ge 3$, $k$-Coloring is NP-hard on $\mathcal{T}_4$. The authors provide ETH-based lower bounds of $2^{\Omega(\sqrt{N})}$ time for these problems by constructing $\varphi$-to-$G(\varphi)$ reductions from $3$-SAT (and Not-All-Equal 3-SAT), with carefully designed variable and clause gadgets that respect the target twin-width. They also discuss structural observations, such as $\mathcal{T}_3$ excluding a fixed planar induced minor, and pose open questions about the limits of algorithmic results on $\mathcal{T}_d$ classes. Overall, the paper significantly advances understanding of which coloring problems remain hard in low-twin-width graph classes and lays groundwork for future structural and algorithmic explorations.

Abstract

As the class $\mathcal T_4$ of graphs of twin-width at most 4 contains every finite subgraph of the infinite grid and every graph obtained by subdividing each edge of an $n$-vertex graph at least $2 \log n$ times, most NP-hard graph problems, like Max Independent Set, Dominating Set, Hamiltonian Cycle, remain so on $\mathcal T_4$. However, Min Coloring and k-Coloring are easy on both families because they are 2-colorable and 3-colorable, respectively. We show that Min Coloring is NP-hard on the class $\mathcal T_3$ of graphs of twin-width at most 3. This is the first hardness result on $\mathcal T_3$ for a problem that is easy on cographs (twin-width 0), on trees (whose twin-width is at most 2), and on unit circular-arc graphs (whose twin-width is at most 3). We also show that for every $k \geqslant 3$, k-Coloring is NP-hard on $\mathcal T_4$. We finally make two observations: (1) there are currently very few problems known to be in P on $\mathcal T_d$ (graphs of twin-width at most $d$) and NP-hard on $\mathcal T_{d+1}$ for some nonnegative integer $d$, and (2) unlike $\mathcal T_4$, which contains every graph as an induced minor, the class $\mathcal T_3$ excludes a fixed planar graph as an induced minor; thus it may be viewed as a special case (or potential counterexample) for conjectures about classes excluding a (planar) induced minor. These observations are accompanied by several open questions.

Coloring Hardness on Low Twin-Width Graphs

TL;DR

This work investigates coloring problems on graphs of bounded twin-width, establishing a sharp hardness boundary: Min Coloring is NP-hard on and, for every fixed , -Coloring is NP-hard on . The authors provide ETH-based lower bounds of time for these problems by constructing -to- reductions from -SAT (and Not-All-Equal 3-SAT), with carefully designed variable and clause gadgets that respect the target twin-width. They also discuss structural observations, such as excluding a fixed planar induced minor, and pose open questions about the limits of algorithmic results on classes. Overall, the paper significantly advances understanding of which coloring problems remain hard in low-twin-width graph classes and lays groundwork for future structural and algorithmic explorations.

Abstract

As the class of graphs of twin-width at most 4 contains every finite subgraph of the infinite grid and every graph obtained by subdividing each edge of an -vertex graph at least times, most NP-hard graph problems, like Max Independent Set, Dominating Set, Hamiltonian Cycle, remain so on . However, Min Coloring and k-Coloring are easy on both families because they are 2-colorable and 3-colorable, respectively. We show that Min Coloring is NP-hard on the class of graphs of twin-width at most 3. This is the first hardness result on for a problem that is easy on cographs (twin-width 0), on trees (whose twin-width is at most 2), and on unit circular-arc graphs (whose twin-width is at most 3). We also show that for every , k-Coloring is NP-hard on . We finally make two observations: (1) there are currently very few problems known to be in P on (graphs of twin-width at most ) and NP-hard on for some nonnegative integer , and (2) unlike , which contains every graph as an induced minor, the class excludes a fixed planar graph as an induced minor; thus it may be viewed as a special case (or potential counterexample) for conjectures about classes excluding a (planar) induced minor. These observations are accompanied by several open questions.
Paper Structure (6 sections, 3 theorems, 1 table)

This paper contains 6 sections, 3 theorems, 1 table.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Standard graph-theoretic definitions and notation
  4. Trigraphs, partition sequences, and twin-width
  5. Hardness of Coloring Graphs of Twin-Width 3
  6. Hardness of 3-Coloring Graphs of Twin-Width 4

Key Result

Theorem 1

Min Coloring is -hard and, unless the ETH fails, requires $2^{\Omega(\sqrt{n})}$ time on $n$-vertex graphs of twin-width at most 3 (even if a 3-sequence is provided).

Theorems & Definitions (3)

  • Theorem 1
  • Theorem 2
  • Corollary 3