Coloring of Graphs Avoiding Bicolored Paths of a Fixed Length

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 \(k=5,6.\)