On graceful difference labelings of disjoint unions of circuits

TL;DR

This paper investigates graceful difference labelings of disjoint unions of circuits, proposing a conjecture that all such graphs, except two cases, admit a gdl, supported by partial proofs.

Contribution

It introduces a conjecture on graceful difference labelings for disjoint unions of circuits and provides partial results supporting it.

Findings

Conjecture that all disjoint unions of circuits have a gdl except two cases

Partial proofs supporting the conjecture

Identification of specific cases where gdl may not exist

Abstract

A graceful difference labeling (gdl for short) of a directed graph G with vertex set V is a bijection f between V and {1,...,|V|} such that, when each arc uv is assigned the difference label f(v)-f(u), the resulting arc labels are distinct. We conjecture that all disjoint unions of circuits have a gdl, except in two particular cases. We prove partial results which support this conjecture.