  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 (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 $P_k$-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 $k\geq4,$
$s_k(G)=O(d^{\frac{k-1}{k-2}}).$ Moreover, we find the exact values for the
$P_k$-chromatic number of the products of some cycles and paths for $k=5,6.$
