Packing spanning rigid subgraphs with restricted degrees

TL;DR

This paper studies how to decompose graphs into spanning subgraphs that are partition-connected and rigid, improving existing connectivity bounds and providing new packing and orientation results.

Contribution

It introduces new conditions for decomposing highly connected graphs into spanning partition-connected and rigid subgraphs, enhancing previous theorems.

Findings

Every sufficiently connected graph contains a spanning subgraph with multiple spanning trees and edge-connected subgraphs.

The results improve bounds on graph connectivity needed for certain decompositions.

Refines conditions for arc-connected orientations of graphs.

Abstract

Let $G$ be a graph and let $l$ be an integer-valued function on subsets of $V(G)$. The graph $G$ is said to be $l$-partition-connected, if for every partition $P$ of $V(G)$, $e_G(P)\ge \sum_{A\in P} l(A)-l(V(G))$, where $e_G(P)$ denotes the number of edges of $G$ joining different parts of $P$. We say that $G$ is $l$-rigid, if it contains a spanning $l$-partition-connected subgraph $H$ with $|E(H)|=\sum_{v\in V(H)} l(v)-l(V(H))$. In this paper, we investigate decomposition of graphs into spanning partition-connected and spanning rigid subgraphs. As a consequence, we improve a recent result due to Gu (2017) by proving that every $(4kp-2p+2m)$-connected graph $G$ with $k\ge 2$ has a spanning subgraph $H$ containing a packing of $m$ spanning trees and $p$ spanning $(2k-1)$-edge-connected subgraphs $H_1,\ldots, H_p$ such that for each vertex $v$, every $H_i-v$ remains $(k-1)$-edge-connected and also $d_H(v)\le \lceil \frac{d_G(v)}{2}\rceil +2kp-p+m$. From this result, we refine a result on arc-connected orientations of graphs.