Acyclic edge coloring of graphs

TL;DR

This paper advances the understanding of acyclic edge coloring by proving the Acyclic Edge Coloring Conjecture for specific classes of graphs, including those with low maximum average degree and certain planar graphs, and establishes exact bounds for particular graph configurations.

Contribution

It provides structural lemmas and proofs confirming the AECC for graphs with maximum average degree less than four, certain planar graphs, and graphs with independent 3+-vertices, extending known results.

Findings

AECC holds for graphs with max average degree less than four

AECC is valid for specific planar graphs without certain triangle adjacencies

Graphs with independent 3+-vertices have acyclic chromatic index equal to maximum degree

Abstract

An {\em acyclic edge coloring} of a graph $G$ is a proper edge coloring such that the subgraph induced by any two color classes is a linear forest (an acyclic graph with maximum degree at most two). The {\em acyclic chromatic index} $\chiup_{a}'(G)$ of a graph $G$ is the least number of colors needed in an acyclic edge coloring of $G$. Fiam\v{c}\'{i}k (1978) conjectured that $\chiup_{a}'(G) \leq \Delta(G) + 2$, where $\Delta(G)$ is the maximum degree of $G$. This conjecture is well known as Acyclic Edge Coloring Conjecture (AECC). A graph $G$ with maximum degree at most $\kappa$ is {\em $\kappa$-deletion-minimal} if $\chiup_{a}'(G) > \kappa$ and $\chiup_{a}'(H) \leq \kappa$ for every proper subgraph $H$ of $G$. The purpose of this paper is to provide many structural lemmas on $\kappa$-deletion-minimal graphs. By using the structural lemmas, we firstly prove that AECC is true for the graphs with maximum average degree less than four (\autoref{NMAD4}). We secondly prove that AECC is true for the planar graphs without triangles adjacent to cycles of length at most four, with an additional condition that every $5$-cycle has at most three edges contained in triangles (\autoref{NoAdjacent}), from which we can conclude some known results as corollaries. We thirdly prove that every planar graph $G$ without intersecting triangles satisfies $\chiup_{a}'(G) \leq \Delta(G) + 3$ (\autoref{NoIntersect}). Finally, we consider one extreme case and prove it: if $G$ is a graph with $\Delta(G) \geq 3$ and all the $3^{+}$-vertices are independent, then $\chiup_{a}'(G) = \Delta(G)$. We hope the structural lemmas will shed some light on the acyclic edge coloring problems.