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

• Published in: Discrete mathematics Vol. 338; no. 10; pp. 1730 - 1739
• Main Authors: Kim, Byeong Moon, Song, Byung Chul, Rho, Yoomi
• Format: Journal Article
• Language: English
• Published: Elsevier B.V 06.10.2015
• Subjects: 2-distance coloring; [formula omitted]-labeling; Cycle; Direct product; Path; Square of a graph; Path; Direct product; L(1,1)-labeling; 2-distance coloring; Cycle; Square of a graph
• ISSN: 0012-365X; 1872-681X
• DOI: 10.1016/j.disc.2014.10.007

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)-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.