An Application
of Graph Pebbling to Zero-Sum
Sequences
in Abelian Groups
Glenn Hurlbert
Department of Mathematics and
Statistics, Arizona State University, USA
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
Erdõs 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 (first
obtained by Geroldinger) in the case of abelian groups. Namely, we
use graph pebbling to prove that for every sequence
of |G| elements of a finite abelian group
G there is a nonempty subsequence 
such that
and 
, where
|g| is the order of the element g 
G.
