Abstract

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.

中文翻译：

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不同类型网格图的两步着色

摘要

在本文中，我们考虑图的两步着色的 NP 困难问题。当没有一对顶点具有相同的颜色时，如果这些顶点彼此之间的距离为一或二，则需要以给定数量的颜色对图进行着色。最佳的两步着色是使用尽可能少的颜色的着色。研究了两步着色问题在网格图中的应用。我们考虑四种类型的网格：三角形、正方形、六边形和八边形。我们证明，在一般情况下，六边形和八边形网格图的最佳两步着色需要四种颜色。我们为这种着色制定了多项式算法。最大顶点度数等于 3 的方形网格图需要四到五种颜色才能进行两步着色。在本文中，我们针对这种情况提出了回溯算法。此外，我们还提出了该算法，该算法以相对于顶点数量的线性时间工作，用于对三角形网格图的七种颜色进行两步着色，并表明这种着色始终是正确的。如果最大顶点度数等于 6，则该解是最优的。