Coloring the Dth Power of The Cartesian Product of Two Cycles and Two Paths

Volume 16, Issue 1, pp 8-17 http://dx.doi.org/10.22436/jmcs.016.01.02

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Authors

Elham Sharifi Yazdi - Department of Computer Engineering, Imam Javad University College, Yazd, Iran.

Abstract

The \(d^{th}\) power graph \(G^d\) is defined on the vertex set of a graph \(G\) in such a way that distinct vertices with distance at most \(d\) in \(G\) are joined by an edge. In this paper the chromatic number of the \(d^{th}\) power of the Cartesian product \(C_m\square C_n\) of two cycles is studied and some of the exact value of \(\chi((C_m\square C_n)^d)\) with conditions are determined. Also the chromatic number of the \(d^{th}\) power of grid \(P_m\square P_n\) with some conditions are determined and the exact value of \(\chi((P_m\square P_n)^d)\) for \(n = 2, 3\) is obtained.

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Keywords

• Chromatic number
• power graphs
• distance \(d\) coloring
• Cartesian product of cycles
• Cartesian product of paths.

MSC

•  05C12
•  05C15
•  05C76

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