Modular chromatic number of $C_m \square P_n$

Document Type : Research Paper

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Annamalai University

Abstract

A modular $k\!$-coloring‎, ‎$k\ge 2,$ of a graph $G$ is a coloring of the vertices of $G$ with the elements in $\mathbb{Z}_k$ having the property that for every two adjacent vertices of $G,$ the sums of the colors of their neighbors are different in $\mathbb{Z}_k.$ The minimum $k$ for which $G$ has a modular $k\!$-coloring is the modular chromatic number of $G.$ Except for some special cases‎, ‎modular chromatic number of $C_m\square P_n$ is determined‎.

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References

R. Balakrishnan and K. Ranganathan (2012). A textbook of graph theory. Second Edition, Universitext. Springer, New York.
F. Okamoto, E. Salehi and P. Zhang (2010). A checkerboard problem and modular colorings of graphs. Bull. Inst. Combin. Appl.. 58, 29-47
F. Okamoto, E. Salehi and P. Zhang (2010). A solution to the checkerboard problem. Int. J. Comput. Appl. Math.. 5, 447-458

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