Spanning trees and spanning closed walks with small degrees

Let G be a graph and let f positive integer-valued function on V(G). In this paper, we show that if for all S⊆V(G), ω(G∖S)<∑v∈S(f(v)−2)+2+ω(G[S]), then has spanning tree T containing an arbitrary given matching such each vertex v, dT(v)≤f(v), where ω(G∖S) denotes the number of components G∖S ω(G[S]) induced subgraph G[S] with set S. This is improvement several results. Next, prove ω(G∖S)≤∑v∈S(f(v)−1)+1, admits closed walk passing through edges meeting v at most f(v) times. result solves long-standing conjecture due to Jackson Wormald (1990).