Download PDFOpen PDF in browserColoring of Graphs Avoiding Bicolored Paths of a Fixed LengthEasyChair Preprint no. 60797 pages•Date: July 13, 2021AbstractThe 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 wellknown examples are star coloring and acyclic coloring of graphs (Gr\"unbaum, 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 {\it $P_k$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 $d\geq 2,$ and $k\geq4,$ $s_k(G)=O(d^{\frac{k1}{k2}}).$ Moreover, we find the exact values for the $P_k$chromatic number of the products of some cycles and paths for $k=5,6.$ Keyphrases: acyclic coloring, graphs, Star coloring
