Let　k　＞=　4　be　an　integer,　and　let　G　be　a　graph　with　maximum　degree　Delta.　In　1991,　Alon,　McDiarmid　and　Reed　proved　that　G　has　a　proper　coloring　using　O(Delta(k-1)/(k-2))　colors　such　that　G　does　not　have　bichromatic　paths　with　k　vertices.　In　this　paper,　we　improve　this　result　by　showing　G　has　a　proper　coloring　using　(1+　left　ceiling　k/2　right　ceiling　1/(k-3))Delta(k-1)/(k-2)+Delta+1colors　such　that　G　does　not　have　bichromatic　paths　with　k　vertices.　We　remark　that　there　exists　a　graph　G　with　maximum　degree　Delta　such　that　for　any　proper　coloring　of　G　using　omega(Delta(k-1)/(k-2)(log　Delta)1/(k-2))　colors,　there　is　always　a　bichromatic　path　with　k　vertices.　Using　the　similar　method,　we　also　show　that　G　has　a　proper　coloring　using　O(Delta　4/3)　colors　such　that　G　contains　neither　bichromatic　cycles　nor　bichromatic　paths　with　order　5.