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：图论中避免子图族中双色成员的顶点着色问题已被广泛研究。最著名的例子包括图的星形着色和无环着色（Grünbaum, 1973），其中分别禁止出现双色的$P_4$副本和双色环。本文研究该问题的一个变体，考虑在网格上进行禁止双色路径的顶点着色。我们将图的$P_k$-色数定义为避免双色$P_k$所需的最少着色顶点数。研究表明，在路径的笛卡尔积$P_{k-2}\square P_{k-2}$的任意3-着色中，必然存在一个双色$P_k$。基于此结果，我们确定了所有$k$值下两条路径乘积（二维网格）的$P_k$-色数问题。