2020 | OriginalPaper | Chapter
List Distinguishing Number of \(p^{\text {th}}\) Power of Hypercube and Cartesian Powers of a Graph
Authors : L. Sunil Chandran, Sajith Padinhatteeri, Karthik Ravi Shankar
Published in: Algorithms and Discrete Applied Mathematics
Publisher: Springer International Publishing
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Abstract
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when a connected graph G is prime with respect to the Cartesian product then \(D_l(G^r) = D(G^r)\) for \(r \ge 3\) where \(G^r\) is the Cartesian product of the graph G taken r times.
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The \(p^{th}\) power of a graph (Some authors use \(G^p\) to denote the pth power of G, to avoid confusion with the notation of Cartesian power of graph G we use \(G^{[p]}\) for the pth power of G.) G is the graph \(G^{[p]}\), whose vertex set is V(G) and in which two vertices are adjacent when they have distance less than or equal to p. We determine \(D_l(Q_n^{[p]})\) for all \(n \ge 7, p \ge 1\), where \(Q_n=K_2^n\) is the hypercube of dimension n.