On constant sum partitions and applications to distance magic-type graphs

Abstract

Let G be an additive abelian group of order n and let n = a 1 + a 2 + ... + a p be a partition of n where 1∑ a∈A i a = t, for some fixed t ∈ G and every 1 ≤ i ≤ p. In 2009, Kaplan, Lev, and Roditty proved that a 0-sum partition of the cyclic group Z n exists for n odd if and only if a 2 ≥ 2. In this paper, we address the case when n is even. In particular, we show that a n 2 -sum partition of Z n exists for n even and p odd if and only if a 2 ≥ 2. Moreover, we provide applications to distance magic-type graphs including the classification of Z n -distance magic complete p-partite graphs for p odd.

Zero-sum partitions of Abelian groups of order $2^n$

“…We only know the following sufficient and necessary condition for Z m -distance magic labeling of a complete t-partite graph only for t or m odd. Theorem 5.1 ( [11,18]). Let G = K n 1 ,n 2 ,...,nt be a complete t-partite graph of order m such that 1 ≤ n 1 ≤ n 2 ≤ .…”