A note on edge-graceful spectra of the square of paths

For a simple path Pr on r vertices, the square of Pr is the graph P2 r on the same set of vertices of Pr , and where every pair of vertices of distance two or less in Pr is connected by an edge. Given a (p, q)-graph G with p vertices and q edges, and a nonnegative integer k, G is said to be k-edge-graceful if the edges can be labeled bijectively by k, k + 1, . . . , k + q − 1, so that the induced vertex sums (mod p) are pairwise distinct, where the vertex sum (mod p) at a vertex is the sum of the labels of all edges incident to such a vertex, modulo the number of vertices p. We call the set of all such k the edge-graceful spectrum of G, and denote it by egI (G). In this article, the edge-graceful spectrum egI (P2 r ) for the square of paths P 2 r is completely determined for odd r . c © 2007 Elsevier B.V. All rights reserved.