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A graph G is H-saturated if it contains no copy of H as a subgraph but the addition of any new edge to G creates a copy of H. In this paper we are interested in the function sat t (n,p), defined to be the minimum number of edges that a Kp -saturated graph on n vertices can have if it has minimum degree at least t. We prove that sat t (n,p) = tn − O(1), where the limit is taken as n tends to infinity. This confirms a conjecture of Bollobás when p = 3. We also present constructions for graphs that give new upper bounds for sat t (n,p).
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