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 \(x \in V(D)\), a subgraph \(H\) of \(D\) is called a \((g, f)\)-factor if \(g(x) \leq d^+_H(x) = d^-_H(x) \leq f(x)\) (\(g(x) \leq d_H(x) \leq f(x)\), resp.) for every \(x \in V(D)\). 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.