Solution of the conjecture: If [formula omitted], [formula omitted], then [formula omitted] has a super vertex-magic total labeling

Abstract

Let 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 and | E | = e . A vertex magic total labeling is a bijection λ from V ∪ E to the consecutive integers 1 , 2 , … , n + e with the property that for every v ∈ V , λ ( v ) + ∑ w ∈ N ( v ) λ ( v , w ) = h , for some constant h. Such a labeling is super if λ ( V ( G ) ) = { 1 , 2 , … , n } . MacDougall, Miller and Sugeng proposed the conjecture: If n ≡ 0 ( mod 4 ) , n > 4 , then K n 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 K 4 l , l > 1 .