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

Sylwia Cichacz

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 $Γ$ is defined as a bijection $\varphi$ $Γ$ such that the mapping $g \mapsto g^{-1}\varphi(g)$ is also bijective. In 1981, Friedlander, Gordon, and Tannenbaum conjectured that when $Γ$ is Abelian, for any $k \geq 2$ dividing $|Γ| -1$, there exists an orthomorphism of $Γ$ 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 $|Γ|\cong 4\pmod{24}$. We also present some applications of zero-sum sets in graph labeling.

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

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 is defined as a bijection such that the mapping is also bijective. In 1981, Friedlander, Gordon, and Tannenbaum conjectured that when is Abelian, for any dividing , there exists an orthomorphism of fixing the identity and permuting the remaining elements as products of disjoint -cycles. Using the idea of disjoint-zero sum subset we provide a solution of this conjecture for and . We also present some applications of zero-sum sets in graph labeling.

Paper Structure

This paper contains 12 sections, 17 theorems, 4 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Zero-sum partition of Abelian group
  4. Skolem partition of Abelian groups
  5. Disjoint zero-sum subsets
  6. Groups orthomorphism
  7. Some applications in graph labeling
  8. Antimagic labeling of trees
  9. Group irregular labeling of graphs
  10. Group irregular labeling of directed graphs
  11. Group distance magic labeling
  12. Statements and Declarations

Key Result

Theorem 3.2

Let $\Gamma$ be a finite Abelian group of order $m$. $\Gamma$ has $2$-ZSPP if and only if $|I(\Gamma)|\in\{0,3\}$.

Theorems & Definitions (24)

  • Definition 3.1: CicrSuch
  • Theorem 3.2: Zeng
  • Theorem 3.3: CaccJiaEgawaTannenbaum2
  • Conjecture 3.4: Tannenbaum1
  • Theorem 3.5: CicrSuch2
  • Theorem 3.6: CicrSuch
  • Theorem 3.7: CicrSuch2
  • Conjecture 3.8: CicZ
  • Theorem 3.9: CicZ
  • Definition 3.10
  • ...and 14 more