Binding Numbers and Connected Factors

In this paper, we obtain some sufficient conditions based on binding number for a graph to have a connected factor with degree restrictions. Let alpha and k be positive integers with alpha + k a parts per thousand yen 4. Let G be a connected graph with a spanning subgraph F, each component of which has order at least alpha. We show that if the binding number of G is greater than (alpha k - alpha)/(alpha k - alpha -1) (k a parts per thousand yen 2) and alpha/(alpha -2) (k = 1) then G has a connected subgraph which has F and in which every vertex v has degree at most deg (F) (v) + k. From the result, we derive various sufficient conditions for a graph to have a connected [a, b]-factor.