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An improvement on the bound for the acyclic chromatic index

Lefteris Kirousis, John Livieratos, Alexandros Singh

Abstract

The acyclic chromatic index (or acyclic edge-chromatic number) of a graph is the least number of colors needed to properly color its edges so that none of its cycles has only two colors. We show that for a graph of max degree $Δ$, the acyclic chromatic index is at most $3.142(Δ-1)+1$, improving on the (best to date) bound of Fialho et al. (2020). Our improvement is made possible by considering unordered (non-plane) trees, instead of ordered (plane) ones, as witness structures for the Lovász Local Lemma, a key combinatorial tool often used in related works. The counting of these witness structures entails methods of Analytic Combinatorics.

An improvement on the bound for the acyclic chromatic index

Abstract

The acyclic chromatic index (or acyclic edge-chromatic number) of a graph is the least number of colors needed to properly color its edges so that none of its cycles has only two colors. We show that for a graph of max degree , the acyclic chromatic index is at most , improving on the (best to date) bound of Fialho et al. (2020). Our improvement is made possible by considering unordered (non-plane) trees, instead of ordered (plane) ones, as witness structures for the Lovász Local Lemma, a key combinatorial tool often used in related works. The counting of these witness structures entails methods of Analytic Combinatorics.
Paper Structure (4 sections, 17 theorems, 42 equations, 2 algorithms)

This paper contains 4 sections, 17 theorems, 42 equations, 2 algorithms.

Table of Contents

  1. Introduction
  2. Notation and terminology
  3. Algorithm EdgeColor
  4. Validation Algorithm

Key Result

Lemma 1

At any step of any successive coloring of the edges of a graph, in any order, there are at most $2 (\Delta-1)$ colors that should be avoided in order to produce a 4-⁠ acyclic coloring.

Theorems & Definitions (35)

  • Definition 1
  • Definition 2
  • Lemma 1: Esperet and Parreau DBLP:journals/ejc/EsperetP13
  • proof : Proof Sketch
  • Remark 1
  • Definition 3
  • Remark 2
  • Lemma 2: Progress Lemma
  • proof
  • Corollary 1
  • ...and 25 more