论文标题
跨越树的连接子图很小
Spanning tree-connected subgraphs with small degrees
论文作者
论文摘要
让$ g $是带有跨度子图$ f $的图形，让$ m $成为一个正整数，让$ f $是$ v（g）$上的正整数值。在本文中，我们表明，如果对于所有$ s \ subseteq v（g）$，$$ω_m（g \ setMinus s）\ le \ sum_ { for each vertex $ v$, $d_H(v)\le f(v)+\max\{0,d_F(v)-m\}$, where $G[S]$ denotes the induced subgraph of $G$ with the vertex set $S$ and $Ω_m(G_0)$ is a parameter to measure $m$-tree-connectivity of a given graph $G_0$. 通过应用此结果，我们表明，每个$ k $连接的图形$ g $带有$ k \ ge 2M $具有跨度$ m $ m $ -tree连接子$ h $ h $，以至于$ d_h（v）\ le \ le \ big \ big \ big \ lceil \ lceil \ frac {m} {m} v（h）$;此外，如果$ g $是$ k $ -tree连接和$ k \ ge m $，则$ g $具有一个跨度$ m $ m $ -tree连接子$ h $ h $，以至于$ d_h（v）\ le \ le \ big \ big \ lceil \ lceil \ frac \ frac {m} v（h）$。结果，我们得出的结论是，每$（2M）$ - 边缘连接的图形与$ r \ ge 4m $允许一个跨度$ m $ m- $ - 树连接的子图，最高学位，最多为300万美元。 接下来，我们证明，图$ g $承认满足$Δ（h）\ le 2m+1 $的跨度$ m $ m- $ -tree连接的子图$ h $，如果对于所有$ s \ subseteq v（g）$，$ $ $ $ $ $ $ $ $ $ $ $ $ $，$ $ $ qub，$ s yminus s）+seetminus s） s）\ le \ frac {1} {m} | s | +1，$$，其中$ω（g \ setMinus s）$和$ iso（g \ setMinus s）$分别表示组件的数量以及分别$ g \ g \ setminus s $的零件数量。结果，我们得出的结论是，每$ m（n-1）$ - 连接$ k_ {1，n} $ - 免费的最低度足够大的简单图形，$ n \ ge 3 $允许最高$ 2M+1 $ $ 2M $ m $ m $ m $ m $ m $ m $ m $ m $ m $ m $ m $ - $ m- $ m-n \ ge 3 $。
Let $G$ be a graph with a spanning subgraph $F$, 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 $S\subseteq V(G)$, $$Ω_m(G\setminus S)\le \sum_{v\in S}\big(f(v)-2m\big)+m+Ω_m(G[S]),$$ then $G$ has a spanning $m$-tree-connected subgraph $H$ containing $F$ such that for each vertex $ v$, $d_H(v)\le f(v)+\max\{0,d_F(v)-m\}$, where $G[S]$ denotes the induced subgraph of $G$ with the vertex set $S$ and $Ω_m(G_0)$ is a parameter to measure $m$-tree-connectivity of a given graph $G_0$. By applying this result, we show that every $k$-edge-connected graph $G$ with $k\ge 2m$ has a spanning $m$-tree-connected subgraph $H$ such that $d_H(v)\le \big\lceil \frac{m}{k}(d_G(v)-2m)\big\rceil+2m$ for each $v\in V(H)$; moreover, if $G$ is $k$-tree-connected and $k\ge m$, then $G$ has a spanning $m$-tree-connected subgraph $H$ such that $d_H(v)\le \big\lceil \frac{m}{k}(d_G(v)-m)\big\rceil+m$ for each $v\in V(H)$. As a consequence, we conclude that every $(r-2m)$-edge-connected graph with $r\ge 4m$ admits a spanning $m$-tree-connected subgraph with maximum degree at most $3m$. Next, we prove that a graph $G$ admits a spanning $m$-tree-connected subgraph $H$ satisfying $Δ(H) \le 2m+1$, if for all $S\subseteq V(G)$, $$ ω(G\setminus S)+\small {\frac{m+1}{2}}\, iso(G\setminus S) \le \frac{1}{m}|S|+1,$$ where $ω(G\setminus S)$ and $iso(G\setminus S)$ denote the number of components and the number of isolated vertices of $G\setminus S$, respectively. As a consequence, we conclude that every $m(n-1)$-connected $K_{1, n}$-free simple graph with a sufficiently large minimum degree and $n\ge 3$ admits a spanning $m$-tree-connected subgraph with maximum degree at most $2m+1$.
