  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 $12$.
