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