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Faster 3-colouring algorithm for graphs of diameter 3

Carla Groenland, Hidde Koerts, Sophie Spirkl

TL;DR

We address 3-coloring on graphs of diameter $3$ and develop a branching algorithm with reductions and a key structural lemma to drive progress. The method extends diameter-$2$ techniques via a 'magic-lemma' that yields well-structured outcomes and enables a final branching rule to reduce the problem size. The algorithm runs in time $2^{O(n^{2/3-\

Abstract

We show that given an $n$-vertex graph $G$ of diameter 3 we can decide if $G$ is $3$-colourable in time $2^{O(n^{2/3-\varepsilon})}$ for any $\varepsilon < 1/33$. This improves on the previous best algorithm of $2^{O((n\log n)^{2/3})}$ from Dębski, Piecyk and Rzążewski [Faster 3-coloring of small-diameter graphs, ESA 2021].

Faster 3-colouring algorithm for graphs of diameter 3

TL;DR

We address 3-coloring on graphs of diameter and develop a branching algorithm with reductions and a key structural lemma to drive progress. The method extends diameter- techniques via a 'magic-lemma' that yields well-structured outcomes and enables a final branching rule to reduce the problem size. The algorithm runs in time $2^{O(n^{2/3-\

Abstract

We show that given an -vertex graph of diameter 3 we can decide if is -colourable in time for any . This improves on the previous best algorithm of from Dębski, Piecyk and Rzążewski [Faster 3-coloring of small-diameter graphs, ESA 2021].
Paper Structure (10 sections, 10 theorems, 25 equations, 1 figure)

This paper contains 10 sections, 10 theorems, 25 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Reduction and branching rules
  4. Main lemma and the last branching rule
  5. Running time analysis
  6. Conclusion
  7. Appendices
  8. Correctness of branching rules B1-B4
  9. Correctness of branching rule B5
  10. Detailed running time calculation

Key Result

Theorem 1

list-3-colouring can be solved in time $2^{O(n^{2/3-\varepsilon})}$ for any fixed $\varepsilon < 1/33$ on the class of $n$-vertex graphs of diameter $3$.

Figures (1)

  • Figure 1: Illustrations of the second and third outcomes in \ref{['lem:magic-lemma']} respectively.

Theorems & Definitions (27)

  • Theorem 1
  • Lemma 2: Chernoff bound
  • Theorem 3: Edwards 2satforcolouring
  • Lemma 4
  • Proof 1: Proof
  • Lemma 5
  • Proof 2: Proof
  • Lemma 6
  • Proof 3: Proof
  • Claim 1
  • ...and 17 more