Extending of edge even graceful labeling of graphs to strong \(r\)-edge even graceful labeling (Q2035720)

Summary: 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 \(\{ 2,4,\dots,2q\}\) such that all the vertex labels \(f^*[ V (G)]\), given by \(f^*(u)=(\sum_{uv \in E (G)} f(uv))\mod (2k)\), 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 \(C_2n\) has a strong \(2\)-edge even graceful labeling when \(n\) is even.