Degree conditions for hamiltonicity: Counting the number of missing edges Original Research Article

In 1960 Ore proved the following theorem: Let G be a graph of order n. If image for every pair of nonadjacent vertices u and image, then G is hamiltonian. In this note we strengthen Oreʹs theorem as follows: we determine the maximum number of pairs of nonadjacent vertices that can have degree sum less than n (i.e. violate Oreʹs condition) but still imply that the graph is hamiltonian. Some other sufficient conditions (i.e. Fanʹs condition) for hamiltonian graphs are strengthened as well.