All graphs have antimagic total labelings

Let G=(V,E) be a finite, simple, indirected and either connected or disconnected. A antimagic total labeling of a graph G=(V,E) with p vertices and q edges is a bijection l:V∪E→{1,2,…,p+q} such that all vertex weights are pairwise distinct, where a vertex weight is the sum of the label of that vertex and labels of all edges incident with the vertex. A graph is antimagic total if it has an antimagic total labeling. An antimagic total labeling l of a graph G=(V,E) is called super (resp. repus) if l(V)={1,2,…,p} (resp. l(V)={q+1,q+2,…,q+p}). A graph is super (resp. repus) antimagic total if it has a super (resp. repus) antimagic total labeling. In this paper we prove that all graphs have antimagic total labelings. We also prove that all graphs have super antimagic total labelings and repus antimagic total labelings. Furthermore, we show that some graphs have super (c,d)-antimagic total labelings and repus (c,d)-antimagic total labelings, that is, labelings that use the p smallest labels (resp. p largest labels) on vertices and, moreover, the vertex weights form arithmetic progression