(untitled) (European Journal of Pure and Applied Mathematics, 2025)
A graph G with p vertices and q edges is said to be edge-graceful if its edges can be labeled from 1 through q, in such a way that the labels induced on the vertices by adding over the labels of incident edges modulo p are distinct. Lo’s Theorem is a known result under this topic, which states that if a graph G with p vertices and q edges is edge-graceful, then p|(q^2+q-p(p-1)2). Since the introduction of the concept of edge-graceful labeling, several works on the edge-graceful labeling of specific families of graphs have been conducted. This paper recalls the definition of water wheel graphs (WW_n) and introduces two new related graphs, the right water wheel graphs (RWW_n) and left water wheel graphs (LWW_n). Next, using Lo’s Theorem, and the concepts of divisibility and Diophantine equations, we proved the non-edge-gracefulness of these graphs. Then, using the same concepts, we determine all the edge-graceful splitting graphs of WW_n, RWW_n, and LWW_n, denoted as S(WW_n), S(RWW_n), and S(LWW_n), respectively. Finally, Python computer programs were created and used to conclude that among these graph families, only S(WW_3), S(RWW_3), and S(LWW_3) are edge-graceful, by providing their corresponding edge-graceful labels.