Let　G　be　a　graph　of　maximum　degree　Δ.　A　proper　vertex　coloring　of　G　is　acyclic　if　there　is　no　bichromatic　cycle.　It　was　proved　by　Alon　et　al.　[Acyclic　coloring　of　graphs.　Random　Structures　Algorithms,　1991,　2(3):　277–288]　that　G　admits　an　acyclic　coloring　with　O(Δ4/3)　colors　and　a　proper　coloring　with　O(Δ(k-1)/(k-2))　colors　such　that　no　path　with　k　vertices　is　bichromatic　for　a　fixed　integer　k　≥　5.　In　this　paper,　we　combine　above　two　colorings　and　show　that　if　k　≥　5　and　G　does　not　contain　cycles　of　length　4,　then　G　admits　an　acyclic　coloring　with　O(Δ(k-1)/(k-2))　colors　such　that　no　path　with　k　vertices　is　bichromatic.　©　2015,　Higher　Education　Press　and　Springer-Verlag　Berlin　Heidelberg.