Some new results about a conjecture by Brian Alspach

Abstract

In this paper, we consider the following conjecture, proposed by Brian Alspach, concerning partial sums in finite cyclic groups: given a subset A of mathbb {Z}_n{setminus } {0} of size k such that sum _{zin A} znot = 0, it is possible to find an ordering (a_1,ldots ,a_k) of the elements of A such that the partial sums s_i=sum _{j=1}^i a_j, i=1,ldots ,k, are nonzero and pairwise distinct. This conjecture is known to be true for subsets of size kle 11 in cyclic groups of prime order. Here, we extend this result to any torsion-free abelian group and, as a consequence, we provide an asymptotic result in mathbb {Z}_n. We also consider a related conjecture, originally proposed by Ronald Graham: given a subset A of mathbb {Z}_p{setminus }{0}, where p is a prime, there exists an ordering of the elements of A such that the partial sums are all distinct. Working with the methods developed by Hicks, Ollis, and Schmitt, based on Alon’s combinatorial Nullstellensatz, we prove the validity of this conjecture for subsets A of size 12.