Packing of Rigid Spanning Subgraphs and Spanning Trees

TL;DR

This paper proves new connectivity conditions under which graphs can be decomposed into rigid and connected spanning subgraphs, generalizing classical results and providing improved bounds for specific properties.

Contribution

It establishes a unified framework for packing rigid and connected spanning subgraphs in highly connected graphs, extending previous theorems and offering transparent proofs.

Findings

Every (6k+2l, 2k)-connected graph contains k rigid and l connected edge-disjoint spanning subgraphs.

Improved bounds: 8-connected graphs pack a spanning tree and a 2-connected subgraph.

14-connected graphs have a 2-connected orientation.

Abstract

We prove that every (6k + 2l, 2k)-connected simple graph contains k rigid and l connected edge-disjoint spanning subgraphs. This implies a theorem of Jackson and Jord\'an [4] and a theorem of Jord\'an [6] on packing of rigid spanning subgraphs. Both these results are generalizations of the classical result of Lov\'asz and Yemini [9] saying that every 6-connected graph is rigid for which our approach provides a transparent proof. Our result also gives two improved upper bounds on the connectivity of graphs that have interesting properties: (1) every 8-connected graph packs a spanning tree and a 2-connected spanning subgraph; (2) every 14-connected graph has a 2-connected orientation.