Distance coloring and distance edge-coloring of d- dimensional lattice

Authors:
Shuang-Liang Tian;Ping Chen
Affiliations:
College of Mathematics and Computer Science, Northwest University for Nationalities, Lanzhou, China;College of Management, Northwest University for Nationalities, Lanzhou, China
Venue:
ICIC'12 Proceedings of the 8th international conference on Intelligent Computing Theories and Applications
Year:
2012

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Abstract

The d -dimensional lattice Ld is the Cartesian product of d two-way infinite paths. k -distance coloring (resp. k-distance edge-coloring) of G is a vertex coloring (resp. edge coloring) of G such that no two vertices (resp. edges) within distance k are given the same color, the minimum number of colors necessary to k -distance color (resp. k -distance edge-color) G, and is denoted by χk(G) (resp. ${\chi}^\prime_{k}(G)$). In this paper, we study the distance coloring and distance edge-coloring of d -dimensional lattice Ld, and give exact value of $\chi_{3}(L_{d}),˜ {\chi}^\prime_{2}(L_{d})˜{\rm and}˜ {\chi}^\prime_{k}(L_{2})$ for any integers d≥2, k≥1.