An application of graph pebbling to zero-sum sequences in abelian groups

TL;DR

This paper applies graph pebbling techniques to prove a conjecture about zero-sum sequences in finite abelian groups, establishing conditions for subsequences summing to zero with bounded reciprocal order sums.

Contribution

It proves a conjecture by Kleitman and Lemke for abelian groups using graph pebbling, linking zero-sum sequences with combinatorial methods.

Findings

Confirmed the conjecture for all finite abelian groups.

Established existence of zero-sum subsequences with bounded reciprocal order sums.

Connected graph pebbling to zero-sum sequence problems.

Abstract

A sequence of elements of a finite group G is called a zero-sum sequence if it sums to the identity of G. The study of zero-sum sequences has a long history with many important applications in number theory and group theory. In 1989 Kleitman and Lemke, and independently Chung, proved a strengthening of a number theoretic conjecture of Erdos and Lemke. Kleitman and Lemke then made more general conjectures for finite groups, strengthening the requirements of zero-sum sequences. In this paper we prove their conjecture in the case of abelian groups. Namely, we use graph pebbling to prove that for every sequence (g_k)_{k=1}^{|G|} of |G| elements of a finite abelian group G there is a nonempty subsequence (g_k)_{k in K} such that sum_{k in K}g_k=0_G and sum_{k in K}1/|g_k|\le 1, where |g| is the order of the element g in G.