On Some Variants of Gracefulness of Cycle Graphs

Abstract

A graph $G=(V(G),E(G))$ is even graceful and equivalently graceful, if there exists an injection $f$ from the set of vertices $V(G)$ to $\{0,1,2,3,4,\dots ,2|E(G)|\}$ such that when each edge $uv$ is assigned the label $|f(u)–f(v)|$, the resulting edge labels are $2,4,6,\dots ,2|E(G)|$. In this work, we use even graceful labeling to give a new proof for necessary and sufficient conditions for the gracefulness of the cycle graph. We extend this technique to odd graceful and super Fibonacci graceful labelings of cycle graphs via some number theoretic concept, called a balanced set of natural numbers.