Edge even graceful labeling of a graph with vertices and edges is a bijective from the set of edge to the set of positive integers such that all the vertex labels , given by , where , 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 - edge even graceful labeling and strong - edge even graceful labeling. We have obtained the necessary conditions for more path-related graphs and cycle-related graphs to be an - edge even graceful graph. Furthermore, the minimum number 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 - edge even graceful labeling was found. Finally, we proved that the even cycle has a strong - edge even graceful labeling when is even.
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