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 α and k be positive integers with α + k ≥ 4. Let G be a connected graph with a spanning subgraph F, each component of which has order at least α. We show that if the binding number of G is greater than (α k − α)/(α k − α −1) (k ≥ 2) and α/(α −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.