Degrees and cycles in digraphs

AbstractIn this article, we give conditions on the total degrees of the vertices in a strong digraph implying the existence of a cycle of length at least ⌈(n−1)h⌉ + 1, where n is the number of vertices of the graph and h an integer, 1⩽h⩽n−1. The same conditions imply the existence of a path of length ⌈(n−1)h⌉ + ⌈(n−2)h⌉. In the case of strong oriented graphs (antisymmetric digraphs) we improve these conditions. In both cases, we show that the given conditions are the best possible