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arXiv 2205.05044math.CO
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具有小度数的生成树连通子图

Spanning tree-connected subgraphs with small degrees

Morteza Hasanvand

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Some removed parts of the former version will be published in some new papers
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AI总结：

本文研究了生成树连通子图的度数约束问题，提出了基于图连通性参数的充分条件，并证明了在特定边连通性和树连通性假设下，图存在具有小最大度的生成m-树连通子图。

摘要：

设 $G$ 为一个具有生成子图 $F$ 的图，设 $m$ 为正整数，并设 $f$ 为 $V(G)$ 上的正整数值函数。本文证明，如果对于所有 $S\subseteq V(G)$，$$\Omega_m(G\setminus S)\le \sum_{v\in S}\big(f(v)-2m\big)+m+\Omega_m(G[S]),$$ 那么 $G$ 包含一个生成 $m$-树连通子图 $H$，它包含 $F$ 且对于每个顶点 $v$，$d_H(v)\le f(v)+\max\{0,d_F(v)-m\\)$，其中 $G[S]$ 表示 $G$ 以顶点集 $S$ 的导出子图，$\Omega_m(G_0)$ 是衡量给定图 $G_0$ 的 $m$-树连通性的参数。应用此结果，我们证明每个 $k$-边连通图 $G$（$k\ge 2m$）包含生成 $m$-树连通子图 $H$，使得对每个 $v\in V(H)$，$d_H(v)\le \lceil \frac{m}{k}(d_G(v)-2m)\rceil+2m$；此外，如果 $G$ 是 $k$-树连通的且 $k\ge m$，则 $G$ 包含生成 $m$-树连通子图 $H$，使得对每个 $v\in V(H)$，$d_H(v)\le \lceil \frac{m}{k}(d_G(v)-m)\rceil+m$。作为推论，我们得出每个 $(r-2m)$-边连通图（$r\ge 4m$）存在最大度至多为 $3m$ 的生成 $m$-树连通子图。接下来，我们证明如果对所有 $S\subseteq V(G)$，$$\omega(G\setminus S)+\frac{m+1}{2}\\, iso(G\setminus S) \le \frac{1}{m}|S|+1,$$ 则图 $G$ 存在满足 $\Delta(H) \le 2m+1$ 的生成 $m$-树连通子图 $H$，其中 $\omega(G\setminus S)$ 和 $iso(G\setminus S)$ 分别表示 $G\setminus S$ 的连通分支数和孤立点数。作为推论，我们得出每个具有足够大最小度且 $n\ge 3$ 的 $m(n-1)$-连通 $K_{1, n}$-free 简单图存在最大度至多为 $2m+1$ 的生成 $m$-树连通子图。

英文摘要：

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 $2m+1$.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

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