Minimal Sum Labeling of Graphs

A graph $G$ is called a sum graph if there is a so-called sum labeling of $G$, i.e. an injective function $\mathrm{\ell }:V(G)\to \mathbb{N}$ 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 $\mathrm{\ell }(u)+\mathrm{\ell }(v)=\mathrm{\ell }(w)$. We say that sum labeling $\mathrm{\ell }$ is minimal if there is a vertex $u\in V(G)$ such that $\mathrm{\ell }(u)=1$. 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.