Spanning trees and spanning closed walks with small degrees

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