Minimal Sum Labeling of Graphs

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 in [6] and [5].

All authors were supported by grant SVV-2017-260452, M. Töpfer and M. Konečný were supported by project CE-ITI P202/12/G061 of GA CR.

The full preprinted version of this paper is available at https://arxiv.org/abs/1708.00552v1.