The zero sum partition of Abelian groups and its applications

Kaplan, Lev and  Roditty [1] introduced a notion of zero-sum partitions of subsets in Abelian groups. Let \Gamma be an Abelian group of order n. We shall say that \Gamma has the zero-sum-partition property (ZSP-property) if every partition n-1 = r_1 + r_2 + \ldots + r_t of n-1, with r_i \geq 2 for 1 \leq i \leq t and for any possible positive integer t, there is a partition of \Gamma – \{0\} into pairwise disjoint subsets A_1, A_2,\ldots , A_t, such that |A_i| = r_i and \sum_{a\in A_i}a = 0 for 1 \leq i \leq t.  They conjectured that every Abelian group \Gamma, which is of odd order or contains exactly three involutions, has the ZSP-property. The conjecture was recently proved by Zeng [2].

In this talk we show some applications of the ZSP-property of groups in some graph labeling problems.

[1] G. Kaplan, A. Lev, Y. Roditty, On zero-sum partitions and anti-magic trees, Discrete Math. 309(8) (2009), 2010–2014.
[2] X. Zeng, On zero-sum partitions of abelian groups. Integers 15 (2015), Paper No. A44, 16 pp.