Spanning tree-connected subgraphs with small degrees

TL;DR

This paper establishes conditions under which a graph contains a spanning m-tree-connected subgraph with bounded degrees, extending classical connectivity and degree bounds to more general settings.

Contribution

It introduces new sufficient conditions involving connectivity and degree parameters for the existence of spanning m-tree-connected subgraphs with degree constraints.

Findings

Every sufficiently edge-connected graph has a spanning m-tree-connected subgraph with degree bounds.

Graphs with certain connectivity and degree conditions admit spanning m-tree-connected subgraphs with maximum degree at most 3m.

Specific conditions involving components and isolated vertices ensure the existence of spanning m-tree-connected subgraphs with degree at most 2m+1.

Abstract

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 $S\subseteq V(G)$, $$\Omega_m(G\setminus S)\le \sum_{v\in S}\big(f(v)-2m\big)+m+\Omega_m(G[S]),$$ then $G$ has a spanning $m$-tree-connected subgraph $H$ containing $F$ 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 $\Omega_m(G_0)$ 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 $\Delta(H) \le 2m+1$, if for all $S\subseteq V(G)$, $$ \omega(G\setminus S)+\small {\frac{m+1}{2}}\, iso(G\setminus S) \le \frac{1}{m}|S|+1,$$ where $\omega(G\setminus S)$ 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 $2m+1$.