Summary: 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 \(k\le 11\) 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 R. L. Graham [in: Proceedings of the Washington State University conference on number theory, March 1971. Pullman, WA: Department of Mathematics and Pi Mu Epsilon, Washington State University. 22–40 (1971; Zbl 0231.10034)]: 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 J. Hicks et al. [J. Comb. Des. 27, No. 6, 369–385 (2019; Zbl 1437.05238)], based on Alon’s combinatorial Nullstellensatz, we prove the validity of this conjecture for subsets \(A\) of size 12.