Highly connected orientations from edge-disjoint rigid subgraphs

TL;DR

This paper proves that highly connected graphs have orientations with high vertex connectivity and contain multiple edge-disjoint rigid subgraphs, advancing understanding of graph connectivity and spanning structures.

Contribution

It establishes that graphs with connectivity proportional to the square of k have k-vertex-connected orientations and contain multiple edge-disjoint rigid subgraphs, confirming long-standing conjectures.

Findings

Graphs with connectivity O(k^2) have k-vertex-connected orientations.

Highly connected graphs contain multiple edge-disjoint rigid spanning subgraphs.

Sufficient connectivity ensures existence of spanning trees with highly connected complements.

Abstract

We give an affirmative answer to a long-standing conjecture of Thomassen, stating that every sufficiently highly connected graph has a $k$-vertex-connected orientation. We prove that a connectivity of order $O(k^2)$ suffices. As a key tool, we show that for every pair of positive integers $d$ and $t$, every $(t \cdot h(d))$-connected graph contains $t$ edge-disjoint $d$-rigid (in particular, $d$-connected) spanning subgraphs, where $h(d) = 10d(d+1)$. This also implies a positive answer to the conjecture of Kriesell that every sufficiently highly connected graph $G$ contains a spanning tree $T$ such that $G-E(T)$ is $k$-connected.