A vertex coloring of a graph G is called r -acyclic if it is a proper vertex coloring such that every cycle C receives at least min{|C|,r} colors.The r -acyclic chromatic number of G is the least number of colors in an r -acyclic coloring of G.We prove that for any number r ≥ 4,the r -acyclic chromatic number of any graph G with maximum degree △ and with girth at least 2(r - 1)△is at most 6(r - 1)△.