Extending of Edge Even Graceful Labeling of Graphs to Strong r-Edge Even Graceful Labeling

Abstract

Edge even graceful labeling of a graph G with p vertices and q edges is a bijective f from the set of edge EG to the set of positive integers 2,4,…,2q such that all the vertex labels f∗VG, given by f∗u=∑uv∈EGfuvmod2k, where k=maxp,q, are pairwise distinct. There are many graphs that do not have edge even graceful labeling, so in this paper, we have extended the definition of edge even graceful labeling to r-edge even graceful labeling and strong r-edge even graceful labeling. We have obtained the necessary conditions for more path-related graphs and cycle-related graphs to be an r-edge even graceful graph. Furthermore, the minimum number r for which the graphs: tortoise graph, double star graph, ladder and diagonal ladder graphs, helm graph, crown graph, sunflower graph, and sunflower planar graph, have an r-edge even graceful labeling was found. Finally, we proved that the even cycle C2n has a strong 2-edge even graceful labeling when n is even.