Graham conjecture on small sets in abelian groups

Abstract

A famous conjecture of Graham asserts that every set $A\subseteq {\mathbb{Z}}_p\setminus \{0\}$ can be ordered so that all partial sums are distinct. Although this conjecture was recently proved for sufficiently large primes by Pham and Sauermann in~\cite{PM} (combined with earlier results of \cite{BBKMM}), it remains open for general abelian groups, even in the cyclic case ${\mathbb{Z}}_k$. In this paper, using a recursive approach, we investigate the sequenceability of subsets $A$ in generic abelian groups for small values of $|A|$. We prove that any subset $A\subseteq G\setminus \{0\}$ with $|A|\le 20$ is sequenceable where previously it was known only for $|A|\le 9$. This bound is improved to $|A|\le 22$ for zero-sum subsets. Finally, regarding the related CMPP conjecture, we show that zero-sum subsets without inverse pairs are sequenceable for $|A|\le 23$.