  Let $G$ be a graph, let $m$ be a positive integer, and let $f$ be a positive
integer-valued function on $V(G)$. In this paper, we show that if for all
$S\subseteq V(G)$, $$\Omega_m(G\setminus S)\le \sum_{v\in
S}\big(f(v)-2m\big)+m+\Omega_m(G[S]),$$ then $G$ has a spanning
$m$-tree-connected subgraph $H$ containing an arbitrary given subgraph with
maximum 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 given
graph $G_0$. By applying this result, we prove that every $k$-edge-connected
graph $G$ has a spanning $m$-tree-connected subgraph $H$ such that for each
vertex $v$, $d_H(v)\le \big\lceil \frac{m}{k}(d_G(v)-2m)\big\rceil+2m$, where
$k\ge 2m$; also if $G$ is $k$-tree-connected, then $H$ can be found such that
for 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 least
three admits a spanning closed trail meeting each vertex at most twice. This
result confirms a weaker version of a long-standing conjecture due to Chv\'atal
(1973).
