Two-Step Coloring of Grid Graphs of Different Types

In this article, we consider the NP-hard problem of the two-step coloring of a graph. It is required to color the graph in the given number of colors in a way, when no pair of vertices has the same color, if these vertices are at a distance of one or two between each other. The optimum two-step coloring is one that uses the minimum possible number of colors. The two-step coloring problem is studied in application to grid graphs. We consider four types of grids: triangular, square, hexagonal, and octagonal. We show that the optimum two-step coloring of hexagonal and octagonal grid graphs requires four colors in the general case. We formulate the polynomial algorithms for such a coloring. A square grid graph with the maximum vertex degree equal to 3 requires four or five colors for a two-step coloring. In this paper, we propose the backtracking algorithm for this case. Also, we present the algorithm, which works in linear time relative to the number of vertices, for the two-step coloring in seven colors of a triangular grid graph and show that this coloring is always correct. If the maximum vertex degree equals six, the solution is optimum.