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 $$\mathbb {Z}_n{\setminus } \{0\}$$ Z n \ { 0 } of size k such that $$\sum _{z\in A} z\not = 0$$ ∑ z ∈ A z ≠ 0 , it is possible to find an ordering $$(a_1,\ldots ,a_k)$$ ( a 1 , … , a k ) of the elements of A such that the partial sums $$s_i=\sum _{j=1}^i a_j$$ s i = ∑ j = 1 i a j , $$i=1,\ldots ,k$$ i = 1 , … , k , are nonzero and pairwise distinct. This conjecture is known to be true for subsets of size $$k\le 11$$ k ≤ 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 $$\mathbb {Z}_n$$ Z n . We also consider a related conjecture, originally proposed by Ronald Graham: given a subset A of $$\mathbb {Z}_p{\setminus }\{0\}$$ Z p \ { 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.

Relative Heffter arrays and biembeddings

“…In the following, if there are no ambiguities, we will just write ω for ω α and ω −1 for ω α −1 omitting the dependence on α. We point out that there are several interesting problems and conjectures about distinct partial sums, see, for instance, [2,6,16,19,23,31]. Given an m × n p.f.…”

On Sequences in Cyclic Groups with Distinct Partial Sums

“…Here we will say that a finite subset S of an abelian group G is non-zero sum if ΣS = 0 G . We recall a definition and lemma from [8]. First, given a subset S of an abelian group G, we define the set Υ(S) = S ∪ ∆(S) ∪ {ΣS} .…”

“…Theorem 1.3 summarizes the main known results concerning Conjecture 1.1. Furthermore, if G is a torsion free abelian group, then any subset S of G \ {0} whose size is at most 11 is sequenceable [8].…”

“…In addition, Theorem 1.4.1 with k = 11 and t = 1 of the present paper was used in [8] as part of a proof that some instances of the conjecture hold when k = 12 and ΣS = 0.…”

Variations on the Erdős distinct-sums problem

“…In this case, we can provide an upper bound by using Alon's combinatorial nullstellensatz. This Theorem has been applied in several Combinatorial Number Theory problems; we refer to [17] (see also [8]) for applications in the similar context of Alspach partial sums conjecture and to [7] for background on that problem. We report here the theorem for the reader's convenience.…”