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Finding large $k$-colorable induced subgraphs in (bull, chair)-free and (bull,E)-free graphs

Nadzieja Hodur, Monika Pilśniak, Magdalena Prorok, Paweł Rzążewski

TL;DR

This work studies Max Partial $k$-Coloring on restricted graph classes, showing sharp algorithmic boundaries for large $k$-colorable induced subgraphs in $(bull,chair)$-free and $(bull,E)$-free graphs. The authors develop a decomposition framework built around the Gyárfás path argument and fat-path/fat-cycle structure to reduce instances to simpler subproblems, enabling recursive solution schemes. Their main results show time $n^{O(k\omega)}$ for $(bull,chair)$-free graphs and $n^{O(k\omega\log n)}$ for $(bull,E)$-free graphs, with immediate corollaries for List-$k$-Coloring and subexponential-time algorithms; a conjecture suggests polynomial-time solvability for the latter class. The paper extends the landscape of tractable cases for coloring-related problems in hereditary graph classes and outlines a broader H-coloring perspective, potentially guiding future complexity results.

Abstract

We study the Max Partial $k$-Coloring problem, where we are given a vertex-weighted graph, and we ask for a maximum-weight induced subgraph that admits a proper $k$-coloring. For $k=1$ this problem coincides with Maximum Weight Independent Set, and for $k=2$ the problem is equivalent (by complementation) to Minimum Odd Cycle Transversal. Furthermore, it generalizes $k$-Coloring. We show that Max Partial $k$-Coloring on $n$-vertex instances with clique number $ω$ can be solved in time * $n^{\mathcal{O}(kω)}$ if the input graph excludes the bull and the chair as an induced subgraph, * $n^{\mathcal{O}(kω\log n)}$ if the input graph excludes the bull and E as an induced subgraph. This implies that $k$-Coloring can be solved in polynomial time in the former class, and in quasipolynomial-time in the latter one.

Finding large $k$-colorable induced subgraphs in (bull, chair)-free and (bull,E)-free graphs

TL;DR

This work studies Max Partial -Coloring on restricted graph classes, showing sharp algorithmic boundaries for large -colorable induced subgraphs in -free and -free graphs. The authors develop a decomposition framework built around the Gyárfás path argument and fat-path/fat-cycle structure to reduce instances to simpler subproblems, enabling recursive solution schemes. Their main results show time for -free graphs and for -free graphs, with immediate corollaries for List--Coloring and subexponential-time algorithms; a conjecture suggests polynomial-time solvability for the latter class. The paper extends the landscape of tractable cases for coloring-related problems in hereditary graph classes and outlines a broader H-coloring perspective, potentially guiding future complexity results.

Abstract

We study the Max Partial -Coloring problem, where we are given a vertex-weighted graph, and we ask for a maximum-weight induced subgraph that admits a proper -coloring. For this problem coincides with Maximum Weight Independent Set, and for the problem is equivalent (by complementation) to Minimum Odd Cycle Transversal. Furthermore, it generalizes -Coloring. We show that Max Partial -Coloring on -vertex instances with clique number can be solved in time * if the input graph excludes the bull and the chair as an induced subgraph, * if the input graph excludes the bull and E as an induced subgraph. This implies that -Coloring can be solved in polynomial time in the former class, and in quasipolynomial-time in the latter one.

Paper Structure

This paper contains 19 sections, 15 theorems, 10 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Graph theory.
  4. Max Partial $k$-Coloring.
  5. Gyárfás path argument.
  6. Auxiliary results
  7. Structural lemmas
  8. Dealing with fat paths and fat cycles
  9. Proof of \ref{['thm:chair', 'thm:E']}
  10. Building the Gyárfás path.
  11. Separating the sets $A_j$ from each other.
  12. Solving the problem on each set $A_j$.
  13. Solving the problem on $R$.
  14. Separating $D$ and $T$.
  15. Solving the problem on $D$ and $T$.
  16. ...and 4 more sections

Key Result

Theorem 1.1

For every $k \geq 1$, Max Partial $k$-Coloring on $(\textrm{bull},\textrm{chair})$-free instances with $n$ vertices and clique number $\omega$ can be solved in time $n^{\mathcal{O}(k \omega)}$.

Figures (6)

  • Figure 1: Considered forbidden subgraphs.
  • Figure 2: Induced subgraphs constructed in the proof of \ref{['clm:NQ']}, the case that $\ell>1$.
  • Figure 3: Induced subgraphs constructed in the proof of \ref{['clm:NQ']}, the case that $\ell=1$.
  • Figure 4: Induced subgraphs constructed in the proof of \ref{['clm:fatpath']}.
  • Figure 5: Induced subgraphs constructed in proofs of \ref{['clm:D_complete_to_R', 'clm:only_D_has_N2', 'clm:bc_comps_compl_to_D']}.
  • ...and 1 more figures

Theorems & Definitions (28)

  • Theorem 1.1
  • Theorem 1.2
  • Corollary 1.2
  • Corollary 1.2
  • Corollary 1.2
  • Conjecture 1.2
  • Theorem 2.1
  • Lemma 3.1
  • proof
  • Claim 3.1.1
  • ...and 18 more