Szczegóły publikacji
Opis bibliograficzny
Zero-sum partitions of Abelian groups of order $2^n$ / Sylwia CICHACZ, Karol SUCHAN // Discrete Mathematics and Theoretical Computer Science ; ISSN 1462-7264. — 2023 — vol. 25 iss. 1 art. no. 6, s. 1–63. — Bibliogr. s. 13–14, Abstr. — K. Suchan - dod. afiliacja: Universidad Diego Portales, Santiago, Chile
Autorzy (2)
Słowa kluczowe
Dane bibliometryczne
| ID BaDAP | 145579 |
|---|---|
| Data dodania do BaDAP | 2024-02-28 |
| Tekst źródłowy | URL |
| DOI | 10.46298/dmtcs.9914 |
| Rok publikacji | 2023 |
| Typ publikacji | artykuł w czasopiśmie |
| Otwarty dostęp |  |
| Creative Commons | |
| Czasopismo/seria | Discrete Mathematics and Theoretical Computer Science |
Abstract
The following problem has been known since the 80's. Let Γ be an Abelian group of order m (denoted |Γ|=m), and let t and mi, 1≤i≤t, be positive integers such that ∑ti=1mi=m−1. Determine when Γ∗=Γ∖{0}, the set of non-zero elements of Γ, can be partitioned into disjoint subsets Si, 1≤i≤t, such that |Si|=mi and ∑s∈Sis=0 for every i, 1≤i≤t. It is easy to check that mi≥2 (for every i, 1≤i≤t) and |I(Γ)|≠1 are necessary conditions for the existence of such partitions, where I(Γ) is the set of involutions of Γ. It was proved that the condition mi≥2 is sufficient if and only if |I(Γ)|∈{0,3}. For other groups (i.e., for which |I(Γ)|≠3 and |I(Γ)|>1), only the case of any group Γ with Γ≅(Z2)n for some positive integer n has been analyzed completely so far, and it was shown independently by several authors that mi≥3 is sufficient in this case. Moreover, recently Cichacz and Tuza proved that, if |Γ| is large enough and |I(Γ)|>1, then mi≥4 is sufficient. In this paper we generalize this result for every Abelian group of order 2n. Namely, we show that the condition mi≥3 is sufficient for Γ such that |I(Γ)|>1 and |Γ|=2n, for every positive integer n. We also present some applications of this result to graph magic- and anti-magic-type labelings.
