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

Abstract

For a simple path P r on r vertices, the square of P r is the graph P r 2 on the same set of vertices of P r , and where every pair of vertices of distance two or less in P r 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 e g I ( G ) . In this article, the edge-graceful spectrum e g I ( P r 2 ) for the square of paths P r 2 is completely determined for odd r .