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(xy) is constant for all edges xy ∈ E. Enomoto, Masuda and Nakamigawa proved that there are magic graphs of order at most 3n + o(n) which contain a complete graph of order n. Bounds on Sidon sets show that the order of such a graph is at least n + o(n). We close the gap between those two bounds by showing that, for any given graph H of order n, there are connected magic graphs of order n + o(n) containing H as an induced subgraph. Moreover it can be required that the graph admits a supermagic labelling f , which satisfies the additional condition f(V ) = [1, |V |].