Abstract
Edge even graceful labeling of a graph G with p vertices and q edges is a bijective f from the set of edge E G to the set of positive integers 2,4 , … , 2 q such that all the vertex labels f ∗ V G , given by f ∗ u = ∑ u v ∈ E G f u v mod 2 k , where k = max p , 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 C 2 n has a strong 2 -edge even graceful labeling when n is even.
Highlights
Introduction andPreliminaries e field of graph theory plays vital role in various mathematical fields which are used in structural models.is structural arrangement of various objects or technologies lead to new inventions and modifications in the existing environment for improvement in these fields
Because there are many graphs which do not have edge even graceful labeling, we introduce the extension of the definition of edge even graceful labeling to r-edge even graceful labeling
It should be noted that the star graph K1,n has edge even graceful labeling only when n is an even number [8]
Summary
2.1. e r-Edge Even Graceful Labeling of the Path Graph Pn. e path graph Pn has edge even graceful labeling only when n is an odd number [8]. It should be noted that the star graph K1,n has edge even graceful labeling only when n is an even number [8]. The comb graph Pn ⊙ K1 has 3-edge even graceful labeling for any number n. E graph Tn,n−2 obtained from a path Pn by attaching exactly one pendant edge to each internal vertex of the path Pn, i.e., Tn,n−2, has vertices {vi, uj, i 1, 2, . E double star Bn,m has edge even graceful labeling when one of (m or n) is an odd number and the other is an even number [11]. (1) e double star Bn,m is 2-edge even graceful graph when both m and n are odd numbers. (2) e double star Bn,m is 3-edge even graceful graph when both m and n are even numbers. 2 4 4 6 6 8 8 10 10 12 12 14 14 16 18 18 Figure 8: A 3-edge even labeling of the graph T9,7
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Free) 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call
