2-distance colorings of some direct products of paths and cycles

The square of a graph is obtained by adding edges between vertices of distance two in the original graph. The 2-distance coloring problem of a graph is the vertex coloring problem of its square graph. Accordingly the chromatic number of 2-distance coloring is called the 2-distance chromatic number. The 2-distance coloring problem is equivalent to a kind of the distance two labeling problem, the L(1,1)L(1,1)-labeling problem which is motivated by the channel assignment problem. In this paper we find the 2-distance chromatic number of the direct product of two cycles whose numbers of vertices are large enough. Moreover we find that also for the direct product of a path and a cycle.