Note on a conjecture of Graham

An old conjecture of Graham stated that if n is a prime and S is a sequence of n terms from the cyclic group C n such that all (nontrivial) zero-sum subsequences have the same length, then S must contain at most two distinct terms. In 1976, Erdős and Szemerédi gave a proof of the conjecture for sufficiently large primes n . However, the proof was complicated enough that the details for small primes were never worked out. Both in the paper of Erdős and Szemerédi and in a later survey by Erdős and Graham, the complexity of the proof was lamented. Recently, a new proof, valid even for non-primes n , was given by Gao, Hamidoune and Wang, using Savchev and Chen’s recently proved structure theorem for zero-sum free sequences of long length in C n . However, as this is a fairly involved result, they did not believe it to be the simple proof sought by Erdős, Graham and Szemerédi. In this paper, we give a short proof of the original conjecture that uses only the Cauchy–Davenport Theorem and pigeonhole principle, thus perhaps qualifying as a simple proof. Replacing the use of the Cauchy–Davenport Theorem with the Devos–Goddyn–Mohar Theorem, we obtain an alternate proof, albeit not as simple, of the non-prime case. Additionally, our method yields an exhaustive list detailing the precise structure of S and works for an arbitrary finite abelian group, though the only non-cyclic group for which the hypotheses are non-void is C 2 ⊕ C 2 m .