A vertex coloring of a graph G is called r-acyclic if it is a proper vertex coloring such that every cycle D receives at least min{|D|,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 △ ≥ 11 and with girth at least (r-1)△ is at most (4r-3)△.