An <i>additive labeling</i> of a graph G is a function ℓ:V(G)→N, such that for every two adjacent vertices v and u of G, Σw∼vℓ(w)≠Σw∼uℓ(w) (x∼y means that x is joined to y). The additive number of G, denoted by η(G), is the minimum number k such that G has a additive labeling ℓ:V(G)→Nk. The additive choosability of a graph G, denoted by ηℓ(G), is the smallest number k such that G has an additive labeling for any assignment of lists of size k to the vertices of G, such that the label of each vertex belongs to its own list. Seamone in his PhD thesis conjectured that for every graph G, η(G)=ηℓ(G). We give a negative answer to this conjecture and we show that for every k there is a graph G such that ηℓ(G)−η(G)≥k. A (0,1)-<i>additive labeling</i> of a graph G is a function ℓ:V(G)→{0,1}, such that for every two adjacent vertices v and u of G, Σw∼vℓ(w)≠Σw∼uℓ(w). A graph may lack any (0,1)-additive labeling. We show that it is NP-complete to decide whether a (0,1)-additive labeling exists for some families of graphs such as perfect graphs and planar triangle-free graphs. For a graph G with some (0,1)-additive labelings, the (0,1)-additive number of G is defined as σ1(G)=minℓ∈ΓΣv∈V(G)ℓ(v) where Γ is the set of (0,1)-additive labelings of G. We prove that given a planar graph that admits a (0,1)-additive labeling, for all ϵ>0 , approximating the (0,1)-additive number within n1−ϵ is NP-hard.