Acyclic chromatic index of triangle-free 1-planar graphs

TL;DR

This paper proves that every triangle-free 1-planar graph can be acyclically edge-colored with at most  more colors than its maximum degree, advancing understanding of coloring properties in such graphs.

Contribution

It establishes an upper bound of  +  colors for acyclic edge coloring of triangle-free 1-planar graphs, a significant step in graph coloring theory.

Findings

Proves  +  color bound for triangle-free 1-planar graphs

Supports the conjecture '  + 2 bound in a specific graph class

Advances understanding of acyclic edge coloring in 1-planar graphs

Abstract

An acyclic edge coloring of a graph $G$ is a proper edge coloring such that every cycle is colored with at least three colors. The acyclic chromatic index $\chiup_{a}'(G)$ of a graph $G$ is the least number of colors in an acyclic edge coloring of $G$. It was conjectured that $\chiup'_{a}(G)\leq \Delta(G) + 2$ for any simple graph $G$ with maximum degree $\Delta(G)$. A graph is {\em $1$-planar} if it can be drawn on the plane such that every edge is crossed by at most one other edge. In this paper, we prove that every triangle-free $1$-planar graph $G$ has an acyclic edge coloring with $\Delta(G) + 16$ colors.