Coloring Graphs Without Bichromatic Cycles or Paths

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.