Edge even and edge odd graceful labelings of Paley Graphs

Edge even graceful labeling is a novel graceful labelling, introduced in 2017 by Elsonbaty and Daoud. A graph G with p vertices and q edges is called an edge even graceful if there is a bijection f: E(G) → {2, 4,. . ., 2q} such that, when each vertex is assigned the sum of the labels of all edges incident to it mod 2k, where k = max (p, q), the resulting vertex labels are distinct. A labeling of G is called edge odd graceful labeling, if there exists a bijection f from the set of edges E(G) to the set {1,3,5,…,2q-1} such that the induced the map f* from the set of vertices V(G) to {0,1,2,.,.,2q-1} given by f*(u) = Σ uv∈E(G) f(uv) (mod 2q) is an injection. A graph which admits edge even (odd) graceful labeling is called an edge even (odd) graceful graph. Paley graphs are dense undirected graphs raised from the vertices as elements of an appropriate finite field by joining pairs of vertices that differ by a quadratic residue. In this paper, we study the construction of edge even (odd) graceful labeling for Paley graphs and prove that Paley graphs of prime order are edge even (odd) graceful.