Closures, cycles, and paths

In 1960 Ore proved the following theorem: Let G be a graph of order n. If d(u)+d(v )≥n for every pair of nonadjacent vertices u and v , then G is hamiltonian. Since then for several other graph properties similar sufficient degree conditions have been obtained, so-called “Ore-type degree conditions”. In [R. J. Faudree, R. H. Schelp, A. Saito, and I. Schiermeyer, Discrete Math 307 (2007), 873–877], Faudree et al. strengthened Ore’s theorem as follows: They determined the maximum number of pairs of nonadjacent vertices that can have degree sum less than n (i.e. violate Ore’s condition) but still imply that the graph is hamiltonian. In this article we prove that for some other graph properties the corresponding Ore-type Journal of Graph Theory 2011 Wiley Periodicals, Inc.