The vertex-coloring problem on graphs avoiding bicolored members of a 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 study a variation of this problem, by considering vertex coloring on grids forbidding bicolored paths. We let $P_k$-chromatic number of a graph be the minimum number of colors needed to color the vertex set properly avoiding a bicolored $P_k.$ We show that in any 3-coloring of the cartesian product of paths, $P_{k-2}\square P_{k-2}$, there is a bicolored $P_k.$ With our result, the problem of finding the $P_k$-chromatic number of product of two paths (2-dimensional grid) is settled for all $k.$

翻译：图论中避免特定子图族出现双色构型的顶点着色问题已被广泛研究。最著名的例子是星着色（Grünbaum, 1973）和无环着色（Grünbaum, 1973），分别禁止出现双色的 $P_4$ 路径和双色环。本文研究该问题的变体，通过考虑禁止双色路径的网格顶点着色问题。我们定义图的 $P_k$-色数为避免双色 $P_k$ 路径所需的最小颜色数。我们证明在路径笛卡尔积 $P_{k-2}\square P_{k-2}$ 的任何 3-着色中，必然存在双色 $P_k$ 路径。结合该结果，二维网格（两条路径的乘积图）的 $P_k$-色数问题对任意 $k$ 值得到了完整解决。