An extension of Nash‐Williams and Tutte's Theorem

The celebrated Nash‐Williams and Tutte's Theorem states that a graph contains edge‐disjoint spanning trees if and only if , where In this paper, we prove that, for integers , , if , then contains edge‐disjoint spanning trees and another forest with , and if is not a spanning tree, then has a component with at least edges. Moreover, the bound of is sharp.