Abstract

The following problem has been known since the 80's. Let ? be an Abelian group of orderm(denoted |?| = m), and let t and mi, 1 ? i ? t, be positive integers such that ?ti=1 mi = m-1. Determine when ?? = ? \ {0}, the set of non-zero elements of ?, can be partitioned into disjoint subsets Si, 1 ? i ? t, such that |Si| = mi and ?s?Si s = 0 for every i ? [1,t]. It is easy to check that mi ? 2 (for every i ? [1,t]) and |I(?)| ? 1 are necessary conditions for the existence of such partitions, where I(?) is the set of involutions of ?. It was proved that the condition mi ? 2 is sufficient if and only if |I(?)| ? {0,3} (see Zeng, (2015)). For other groups (i.e., for which |I(?)| ? 3 and |I(?)| > 1), only the case of any group ? with ? ? (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 |?| is large enough and |I(?)| > 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 ? such that |I(?)| > 1 and |?| = 2n, for every positive integer n. We also present some applications of this result to graph magic-and anti-magic-type labelings. © 2023 by the author(s).