Solution of the conjecture: If n≡0(mod4), n>4n>4, then KnKn has a super vertex-magic total labeling

Let G=(V,E)G=(V,E) be a finite non-empty graph, where V and E are the sets of vertices and edges of G  , respectively, and |V|=n|V|=n and |E|=e|E|=e. A vertex magic total labeling is a bijection λλ from V∪EV∪E to the consecutive integers 1,2,…,n+e1,2,…,n+e with the property that for every v∈Vv∈V, λ(v)+∑w∈N(v)λ(v,w)=hλ(v)+∑w∈N(v)λ(v,w)=h, for some constant h  . Such a labeling is super if λ(V(G))={1,2,…,n}λ(V(G))={1,2,…,n}. MacDougall, Miller and Sugeng proposed the conjecture: If n≡0(mod4), n>4n>4, then KnKn has a super vertex-magic total labeling (VMTL). We prove that this conjecture holds true by means of giving a family of super VMTLs of K4lK4l, l>1l>1.