Extensions of a theorem of Erdos on nonhamiltonian graphs

Let n, d be integers with 1 <= d <= left perpendicular n-1/2 right perpendicular, and set h(n,d) := ((n-d)(2)) + d(2). Erdos proved that when n >= 6d, each n-vertex nonhamiltonian graph G with minimum degree delta(G) >= d has at most h(n, d) edges. He also provides a sharpness example H-n,H-d for all such pairs (n, d). Previously, we showed a stability version of this result: for n large enough, every nonhamiltonian graph G on n vertices with delta(G) >= d and more than h(n, d + 1) edges is a subgraph of H-n,H-d. In this article, we show that not only does the graph H-n,H-d maximize the number of edges among nonhamiltonian graphs with n vertices and minimum degree at least d, but in fact it maximizes the number of copies of any fixed graph F when n is sufficiently large in comparison with d and vertical bar F vertical bar. We also show a stronger stability theorem, that is, we classify all nonhamiltonian n-vertex graphs with delta(G) >= d and more than h (n, d + 2) edges. We show this by proving a more general theorem: we describe all such graphs with more than ((n-(d+2))(k)) + (d + 2) ((d+2)(k-1)) copies of K-k for any k.