Let　k≥　4　be　an　integer,　and　let　G　be　a　graph　with　maximum　degree　Δ.　In　1991,　Alon,　McDiarmid　and　Reed　proved　that　G　has　a　proper　coloring　using　O(Δ　(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　+　⌈　k/　2　⌉　1/(k-3))　Δ　(k-1)/(k-2)+　Δ　+　1　colors　such　that　G　does　not　have　bichromatic　paths　with　k　vertices.　We　remark　that　there　exists　a　graph　G　with　maximum　degree　Δ　such　that　for　any　proper　coloring　of　G　using　Ω(Δ(k-1)/(k-2)(logΔ)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(Δ　4　/　3)　colors　such　that　G　contains　neither　bichromatic　cycles　nor　bichromatic　paths　with　order　5.　©　2020,　Malaysian　Mathematical　Sciences　Society　and　Penerbit　Universiti　Sains　Malaysia.