Coloring of Graphs Avoiding Bicolored Paths of a Fixed Length

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