A Generalization of Graham’s Conjecture

Huanhuan Guan,Pingzhi Yuan,Xiangneng Zeng

doi:10.37236/5189

Abstract

Let $G$ be a finite Abelian group of order $|G|=n$, and let $S=g_1\cdot\ldots\cdot g_{n-1}$ be a sequence over $G$ such that all nonempty zero-sum subsequences of $S$ have the same length. In this paper, we completely determine the structure of these sequences.

Highlights

Let G be a finite Abelian group, and let F(G) denote the free Abelian monoid with basis G, the elements of which are called sequences
We say that S contains some g ∈ G if vg(S) 1, and a sequence T ∈ F(G) is a subsequence of S if vg(T ) vg(S) for every g ∈ G, denoted T |S
If all nontrivial zero-sum subsequences of S are of the same length, the number of distinct terms in S is at most 2

Summary

Introduction

Let G be a finite Abelian group (written additively), and let F(G) denote the free Abelian monoid with basis G, the elements of which are called sequences (over G). If all nontrivial zero-sum subsequences of S are of the same length, the number of distinct terms in S is at most 2. Grynkiewicz [8] gave an exhaustive list detailing the precise structure of S and showed that the result holds in an arbitrary finite Abelian group The main purpose of the present paper is to give an exhaustive list detailing the precise structure of the sequences for a slight generalization of Graham’s Conjecture without using the Devos-Goddyn-Mohar Theorem. Suppose that all nontrivial zero-sum subsequences of S have the same length r ∈ [1, n − 1], S is one of the following:. If there is an integer r 0 such that all nonempty zero-sum subsequences of S have length r, |supp(Sx−1)| 2 for some x ∈ supp(S)

Preliminaries

The proof of the main results