Largest cliques in connected supermagic graphs

A graph G = ( V , E ) is said to be magic if there exists an integer labeling f : V ∪ E ⟶ [ 1 , | V ∪ E | ] such that f ( x ) + f ( y ) + f ( x y ) is constant for all edges x y ∈ E . o, Masuda and Nakamigawa proved that there are magic graphs of order at most 3 n 2 + o ( n 2 ) which contain a complete graph of order n . Bounds on Sidon sets show that the order of such a graph is at least n 2 + o ( n 2 ) . We close the gap between those two bounds by showing that, for any given connected graph H of order n , there is a connected magic graph G of order n 2 + o ( n 2 ) containing H as an induced subgraph. Moreover G admits a supermagic labeling f , which satisfies the additional condition f ( V ) = [ 1 , | V | ] .