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arXiv · 1504.06234

Acyclic chromatic index of triangle-free 1-planar graphs

Jijuan Chen·Tao Wang·Huiqin Zhang
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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.

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Jijuan Chen, Tao Wang, Huiqin Zhang. 2015-04-23. Acyclic chromatic index of triangle-free 1-planar graphs. https://doi.org/10.1007/s00373-017-1809-0

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