Graham conjecture on small sets in abelian groups

TL;DR

This paper advances understanding of Graham's conjecture by proving that small subsets in generic abelian groups are sequenceable, extending known bounds from 9 to 22 elements, and addressing related zero-sum subset conjectures.

Contribution

It improves bounds on the size of sequenceable subsets in abelian groups and explores the sequenceability of zero-sum subsets without inverse pairs.

Findings

Any subset with up to 20 elements is sequenceable.

Zero-sum subsets with up to 22 elements are sequenceable.

Zero-sum subsets without inverse pairs are sequenceable up to 23 elements.

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 $|A| \leq 20$ is sequenceable where previously it was known only for $|A|\leq 9$. This bound is improved to $|A| \leq 22$ for zero-sum subsets. Finally, regarding the related CMPP conjecture, we show that zero-sum subsets without inverse pairs are sequenceable for $|A| \leq 23$.