The star chromatic number on a graph is the minimum number of colors in a proper vertex coloring forbidding any $P_4$ with two colors (bicolored). This problem was introduced by Grünbaum (1973) together with the acyclic coloring of graphs, where bicolored cycles are avoided. In this paper, we study a generalization of this problem, by considering proper vertex coloring on graphs forbidding bicolored paths of a fixed length, which was initially discussed by Alon, McDiarmid, and Reed (1991). Here, we study this problem on products of two paths. We show that at least 4 colors are needed to properly color the product of paths, $P_m\square P_n$, avoiding a bicolored $P_k,$ unless $n<k-2$ or $m<k-2.$ With this result, the above question is settled for all $k$ on 2-dimensional grids.

翻译：图的星色数是顶点正常染色所需的最小颜色数，且要求禁止任何由两种颜色（双色）构成的$P_4$。该问题由Grünbaum（1973）与图的非循环染色（避免双色圈）同时提出。本文研究这一问题的推广形式，即在图的顶点正常染色中禁止固定长度的双色路径，该变体最初由Alon、McDiarmid和Reed（1991）讨论。我们聚焦于两条路径的笛卡尔积图上的情形。结果表明：对于路径的乘积图$P_m\square P_n$，为避免双色$P_k$，至少需要4种颜色进行正常染色，除非$n<k-2$或$m<k-2$。该结论解决了二维网格上所有$k$值的上述问题。