Abstract

The following problem has been known since the 80's. Let Γ be an Abelian group of orderm(denoted |Γ| = m), and let t and mi, 1 ≤ i ≤ t, be positive integers such that Σti=1 mi = 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 Σs∈Si s = 0 for every i ∈ [1,t]. It is easy to check that mi ≥ 2 (for every i ∈ [1,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} (see Zeng, (2015)). 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 (2021) 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.