Connected (g, f)-factors and supereulerian digraphs

Given a digraph (an undirected graph, resp.) D and two positive integers f (x); g(x) for every x 2 V (D), a subgraph H of D is called a (g; f)-factor if g(x) d + H (x) = d ? H (x) f (x)(g(x) d H (x) f (x), resp.) for every x 2 V (D). If f (x) = g(x) = 1 for every x, then a connected (g; f)-factor is a hamiltonian cycle. The previous research related to the topic has been carried out either for (g; f)-factors (in general, disconnected) or for hamiltonian cycles separately, even though numerous similarities between them have been recently detected. Here we consider connected (g; f)-factors in digraphs and show that several results on hamiltonian digraphs, which are generalizations of tournaments, can be extended to connected (g; f)-factors. Applications of these results to supereulerian digraphs are also obtained.