Largest cliques in connected supermagic graphs

A graph G=(V,E)G=(V,E) is said to be magic   if there exists an integer labeling f:V∪E⟶[1,|V∪E|]f:V∪E⟶[1,|V∪E|] such that f(x)+f(y)+f(xy)f(x)+f(y)+f(xy) is constant for all edges xy∈Exy∈E.Enomoto, Masuda and Nakamigawa proved that there are magic graphs of order at most 3n2+o(n2)3n2+o(n2) which contain a complete graph of order nn. Bounds on Sidon sets show that the order of such a graph is at least n2+o(n2)n2+o(n2). We close the gap between those two bounds by showing that, for any given connected graph HH of order nn, there is a connected magic graph GG of order n2+o(n2)n2+o(n2) containing HH as an induced subgraph. Moreover GG admits a supermagic labeling ff, which satisfies the additional condition f(V)=[1,|V|]f(V)=[1,|V|].