Coloring the Dth Power of The Cartesian Product of Two Cycles and Two Paths
Elham Sharifi Yazdi · Journal of Mathematics and Computer Science · 2016
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.