On Edge-Graceful Regular Graphs with Particular 3-Factors

An edge-graceful labeling of a finite simple graph with p vertices and q edges is a bijection from the set of edges to the set of integers {1,2,⋯,q} such that the vertex sums are pairwise distinct modulo p, where the vertex sum at a vertex is the sum of labels of all edges incident to such vertex. A graph is called edge-graceful if it admits an edge-graceful labeling. In this article, we verify that an regular graph of odd degree is edge-graceful if it contains either of two particular 3-regular spanning subgraphs, namely, a quasi-prism factor and a claw factor.