Article

# Acyclic edge coloring of sparse graphs

02/2012;
Source: arXiv

ABSTRACT A proper edge coloring of a graph $G$ is called acyclic if there is no
bichromatic cycle in $G$. The acyclic chromatic index of $G$, denoted by
$\chi'_a(G)$, is the least number of colors $k$ such that $G$ has an acyclic
edge $k$-coloring. The maximum average degree of a graph $G$, denoted by
$\mad(G)$, is the maximum of the average degree of all subgraphs of $G$. In
this paper, it is proved that if $\mad(G)<4$, then
$\chi'_a(G)\leq{\Delta(G)+2}$; if $\mad(G)<3$, then
$\chi'_a(G)\leq{\Delta(G)+1}$. This implies that every triangle-free planar
graph $G$ is acyclically edge $(\Delta(G)+2)$-colorable.

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20 Apr 2013

### Keywords

acyclic chromatic index

average degree

bichromatic cycle

denoted

graph $G$

maximum average degree

proper edge coloring