Coloring Grids Avoiding Bicolored Paths

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 Gr\"unbaum (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.