Extensions of a theorem of Erdős on nonhamiltonian graphs

Abstract

AbstractLet be integers with , and set . Erdős proved that when , each n‐vertex nonhamiltonian graph G with minimum degree has at most edges. He also provides a sharpness example for all such pairs . Previously, we showed a stability version of this result: for n large enough, every nonhamiltonian graph G on n vertices with and more than edges is a subgraph of . In this article, we show that not only does the graph 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 . We also show a stronger stability theorem, that is, we classify all nonhamiltonian n‐vertex graphs with and more than edges. We show this by proving a more general theorem: we describe all such graphs with more than copies of for any k.