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A quadratic-time coloring algorithm for graphs with large maximum degree

Feng Liu, Shuang Sun, Yan Wang

Abstract

Graph coloring is a central problem in graph theory and is NP-hard for general graphs. Motivated by the Borodin--Kostochka conjecture, we study the algorithmic problem of coloring graphs with large maximum degree and no clique of size $Δ$. We give a quadratic-time coloring algorithm that constructs a $(Δ-1)$-coloring for such graphs. We also prove that every graph $G$ with maximum degree $Δ\ge 7.3 \times 10^9$ and clique number $ω(G) < Δ$ satisfies $χ(G) \le Δ- 1$. This improves a longstanding result of Reed.

A quadratic-time coloring algorithm for graphs with large maximum degree

Abstract

Graph coloring is a central problem in graph theory and is NP-hard for general graphs. Motivated by the Borodin--Kostochka conjecture, we study the algorithmic problem of coloring graphs with large maximum degree and no clique of size . We give a quadratic-time coloring algorithm that constructs a -coloring for such graphs. We also prove that every graph with maximum degree and clique number satisfies . This improves a longstanding result of Reed.
Paper Structure (7 sections, 19 theorems, 57 equations, 1 algorithm)

This paper contains 7 sections, 19 theorems, 57 equations, 1 algorithm.

Table of Contents

  1. Introduction
  2. Structural decomposition
  3. A proper coloring
  4. Ensuring two repeated colors in $N(v)$ via Azuma and the Local Lemma
  5. Bounding $\mathbb{P}(E_i)$ for near-cliques
  6. Estimating $F_i$ and a lower bound for $\Delta$
  7. A quadratic-time coloring algorithm

Key Result

Theorem 1.1

If $G$ is a connected graph with $\Delta(G)\ge 3$, then $\chi(G)\le \max\{\Delta(G),\omega(G)\}$.

Theorems & Definitions (27)

  • Theorem 1.1: Brooks Brooks1941
  • Conjecture 1.2: Borodin-Kostochka Borodin-Kostochka1977
  • Theorem 1.3
  • Theorem 1.4
  • Lemma 2.1: Reed Reed1999
  • Lemma 2.2: Reed Reed1999
  • Lemma 2.3
  • proof
  • Lemma 2.4
  • Lemma 2.5
  • ...and 17 more