Edge-graceful usual fan graphs
Aaron D. C. Angel, John Rafael M. Antalan, John Loureynz F. Gamurot, Richard P. Tagle
TL;DR
This work addresses the problem of identifying edge-graceful labelings for usual fan graphs $F_{1,n}$. It combines Lo's Theorem, divisibility arguments, and reductions of quadratic Diophantine equations to constrain feasible $n$ values, then uses a computer search to construct explicit edge-graceful labelings. The authors prove that only $F_{1,2}$, $F_{1,3}$, and $F_{1,11}$ are edge-graceful among the usual fans, providing concrete labelings for each. The study demonstrates a practical, computer-assisted approach that integrates graph labeling theory with number-theoretic techniques, and suggests extending the methodology to other graph families in future work.
Abstract
A graph $G$ with $p$ vertices and $q$ edges is said to be edge-graceful if its edges can be labeled from $1$ through $q$, in such a way that the labels induced on the vertices by adding over the labels of incident edges modulo $p$ are distinct. A known result under this topic is Lo's Theorem, which states that if a graph $G$ with $p$ vertices and $q$ edges is edge-graceful, then $p\Big|\Big(q^{2}+q-\dfrac{p(p-1)}{2}\Big)$. This paper presents novel results on the edge-gracefulness of the usual fan graphs. Using Lo's Theorem, the concepts of divisibility and Diophantine equations, and a computer program created, we determine all edge-graceful usual fan graphs $F_{1,n}$ with their corresponding edge-graceful labels.