Coloring of Graphs Avoiding Bicolored Paths of a Fixed Length

Abstract

The problem of finding the minimum number of colors to color a graph properly without containing any bicolored copy of a fixed family of subgraphs has been widely studied. Most well-known examples are star coloring and acyclic coloring of graphs (Grünbaum in Isreal J Math 14(4):390–498, 1973) where bicolored copies of $P_4$ and cycles are not allowed, respectively. In this paper, we introduce a variation of these problems and study proper coloring of graphs not containing a bicolored path of a fixed length and provide general bounds for all graphs. A $P_k$-coloring of an undirected graph G is a proper vertex coloring of G such that there is no bicolored copy of $P_k$ in G,  and the minimum number of colors needed for a $P_k$-coloring of G is called the $P_k$-chromatic number of G,  denoted by $s_k(G).$ We provide bounds on $s_k(G)$ for all graphs, in particular, proving that for any graph G with maximum degree $d\ge 2,$ and $k\ge 4,$ $s_k(G)\le \lceil 6\sqrt{10}d^{\frac{k-1}{k-2}} \rceil .$ Moreover, we find the exact values for the $P_k$-chromatic number of the products of some cycles and paths for $k=5,6.$