Zero-sum partitions of Abelian groups and their applications to magic-type labelings

The following problem has been known since the 80s. Let Γ be an Abelian group of order m (denoted |Γ|=m), and let t and {m_i}_i=1^t, be positive integers such that ∑_i=1^t m_i=m-1. Determine when Γ^*=Γ∖{0}, the set of non-zero elements of Γ, can be partitioned into disjoint subsets {S_i}_i=1^t such that |S_i|=m_i and ∑_s∈ S_is=0 for every 1 ≤ i ≤ t. Such a subset partition is called a zero-sum partition. |I(Γ)|≠ 1, where I(Γ) is the set of involutions in Γ, is a necessary condition for the existence of zero-sum partitions. In this paper, we show that the additional condition of m_i≥ 4 for every 1 ≤ i ≤ t, is sufficient. Moreover, we present some applications of zero-sum partitions to magic-type labelings of graphs.