A zero-sum problem on graphs

TL;DR

This paper investigates which graphs guarantee a zero-sum subset for any labeling with a finite abelian group, revealing a complete characterization for prime order groups and partial results for composite groups.

Contribution

It characterizes zero-forcing graphs for cyclic groups of prime order and explores structural conditions and computational aspects for general finite abelian groups.

Findings

Connected graphs with at least p vertices are zero-forcing for cyclic groups of prime order p.

Zero-forcing property depends on the structure of the graph when the group order is composite.

Partial solutions and open questions are presented for the general case.

Abstract

Call a graph $G$ zero-forcing for a finite abelian group $\mathcal{G}$ if for every $\ell : V(G) \to \mathcal{G}$ there is a connected $A \subseteq V(G)$ with $\sum_{a \in A} \ell(a) = 0$. The problem we pose here is to characterise the class of zero-forcing graphs. It is shown that a connected graph is zero-forcing for the cyclic group of prime order $p$ if and only if it has at least $p$ vertices. When $|\mathcal{G}|$ is not prime, however, being zero-forcing is intimately linked to the structure of the graph. We obtain partial solutions for the general case, discuss computational issues and present several questions.