Disjoint zero-sum subsets in Abelian groups and its application -- survey

TL;DR

This survey summarizes research on disjoint zero-sum subsets in finite Abelian groups, highlighting recent solutions to conjectures and applications in graph labeling within additive and combinatorial number theory.

Contribution

It provides a solution to a conjecture on orthomorphisms for specific group orders and explores applications of zero-sum sets in graph labeling.

Findings

Solved the conjecture for k=3 and groups with order ≡ 4 mod 24.

Connected zero-sum sets to orthomorphisms in Abelian groups.

Demonstrated applications in graph labeling.

Abstract

We provide a summary of research on disjoint zero-sum subsets in finite Abelian groups, which is a branch of additive group theory and combinatorial number theory. An orthomorphism of a group $\Gamma$ is defined as a bijection $\varphi$ $\Gamma$ such that the mapping $g \mapsto g^{-1}\varphi(g)$ is also bijective. In 1981, Friedlander, Gordon, and Tannenbaum conjectured that when $\Gamma$ is Abelian, for any $k \geq 2$ dividing $|\Gamma| -1$, there exists an orthomorphism of $\Gamma$ fixing the identity and permuting the remaining elements as products of disjoint $k$-cycles. Using the idea of disjoint-zero sum subset we provide a solution of this conjecture for $k=3$ and $|\Gamma|\cong 4\pmod{24}$. We also present some applications of zero-sum sets in graph labeling.