A proper edge coloring of a graph is acyclic, if every cycle of the graph has at least 3 colors. Let
r be a positive integer. An edge coloring is
r-acyclic if it is proper and every cycle
C has at least

colors. The
r-acyclic edge chromatic

number of a graph
G is the minimum number of colors needed for any
r-acyclic edge coloring of
G. When r=4, the result of this paper is that the 4-acyclic chromatic number of a graph with maximum degree Δ and girth

is less than 18Δ. Furthermore, if the girth of graph
G is at least

, then

.