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Let $G$ be a graph, let $m$ be a positive integer, and let $f$ be a positiveinteger-valued function on $V(G)$. In this paper, we show that if for all$S\subseteq V(G)$, $$\Omega_m(G\setminus S)\le \sum_{v\inS}\big(f(v)-2m\big)+m+\Omega_m(G[S]),$$ then $G$ has a spanning$m$-tree-connected subgraph $H$ containing an arbitrary given subgraph withmaximum degree at most $m$ such that for each vertex $v$, $d_H(v)\le f(v)$,where $G[S]$ denotes the induced subgraph of $G$ with the vertex set $S$ and$\Omega_m(G_0)$ is a parameter to measure $m$-tree-connectivity of a givengraph $G_0$. By applying this result, we prove that every $k$-edge-connectedgraph $G$ has a spanning $m$-tree-connected subgraph $H$ such that for eachvertex $v$, $d_H(v)\le \big\lceil \frac{m}{k}(d_G(v)-2m)\big\rceil+2m$, where$k\ge 2m$; also if $G$ is $k$-tree-connected, then $H$ can be found such thatfor each vertex $v$, $d_H(v)\le \big\lceil \frac{m}{k}(d_G(v)-m)\big\rceil+m$,where $k\ge m$. Next, we prove that if for all $S\subseteq V(G)$,$$\frac{1}{2}\Omega_2(G\setminus S)\le \sum_{v\in S}(f(v)-3/2)+1,$$ then $G$admits a spanning closed trail meeting each vertex $v$ at most $f(v)$ times.From this result, we conclude that every $4$-tough graph of order at leastthree admits a spanning closed trail meeting each vertex at most twice. Thisresult confirms a weaker version of a long-standing conjecture due to Chv\'atal(1973).

Spanning tree-connected subgraphs and spanning closed trails with small degrees