Edge-graceful usual fan graphs

TL;DR

This paper investigates the edge-gracefulness of usual fan graphs, applying Lo's Theorem and Diophantine equations, and provides a complete characterization with explicit labels for these graphs.

Contribution

It offers new results on the edge-gracefulness of usual fan graphs, including a complete classification and explicit labeling using computational methods.

Findings

Identifies all edge-graceful usual fan graphs $F_{1,n}$.

Provides explicit edge-labelings for these graphs.

Extends understanding of graph labelings through computational verification.

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.