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Some new results about a conjecture by Brian Alspach

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 $\sum_{z\in A} z\not= 0$, 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 $k\leq 11$ in cyclic groups of prime order. Here, we extend such 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 the Alon's combinatorial Nullstellensatz, we prove the validity of such conjecture for subsets $A$ of size $12$.