Abstract
A graph G is called a sum graph if there is a sum labeling of G, i.e., an injective function ℓ:V(G)→N such that for every u,v∈V(G) it holds that uv∈E(G) if and only if there exists a vertex w∈V(G) such that ℓ(u)+ℓ(v)=ℓ(w). We say that sum labeling ℓ is minimal if there is a vertex u∈V(G) such that ℓ(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.
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