Minimal Sum Labeling of Graphs

TL;DR

This paper investigates the properties of sum graphs, showing that relaxing certain conditions leads to the existence of sum graphs without minimal labelings, addressing a question from 1998.

Contribution

It demonstrates that relaxing injectivity or loop conditions in sum graphs can result in graphs lacking minimal labelings, answering a long-standing open question.

Findings

Sum graphs without minimal labelings exist under relaxed conditions.

Relaxing injectivity or allowing loops affects the existence of minimal labelings.

Partially answers the question posed by Miller, Ryan, and Smyth in 1998.

Abstract

A graph $G$ is called a sum graph if there is a so-called sum labeling of $G$, i.e. an injective function $\ell: V(G) \rightarrow \mathbb{N}$ such that for every $u,v\in V(G)$ it holds that $uv\in E(G)$ if and only if there exists a vertex $w\in V(G)$ such that $\ell(u)+\ell(v) = \ell(w)$. We say that sum labeling $\ell$ is minimal if there is a vertex $u\in V(G)$ such that $\ell(u)=1$. In this paper, we show that if we relax the conditions (either allow non-injective labelings or consider graphs with loops) then there are sum graphs without a minimal labeling, which partially answers the question posed by Miller, Ryan and Smyth in 1998.