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Highly connected orientations from edge-disjoint rigid subgraphs

Dániel Garamvölgyi, Tibor Jordán, Csaba Király, Soma Villányi

TL;DR

This work settles Thomassen's conjecture on the existence of $k$-connected orientations in sufficiently highly connected graphs, proving a quadratic connectivity sufficiency bound and introducing a powerful packing theorem for rigid spanning subgraphs. Central to the approach is the $d$-dimensional rigidity matroid and its unions, enabling the construction of $t$ edge-disjoint $d$-rigid spanning subgraphs under high connectivity. By combining two edge-disjoint minimally $d$-rigid subgraphs with carefully chosen orientations (via a $(d,R)$-orientation) and applying Hakimi's orientation criterion, the authors obtain $k$-connected orientations under explicit connectivity. They further adapt these ideas to Kriesell's spanning-tree conjecture, showing that sufficiently connected graphs contain a spanning tree whose removal leaves a $k$-connected graph, with explicit bounds. The results are supported by probabilistic methods, combinatorial matroid theory, and yield algorithmic pathways for constructing the desired packings and orientations.

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.

Highly connected orientations from edge-disjoint rigid subgraphs

TL;DR

This work settles Thomassen's conjecture on the existence of -connected orientations in sufficiently highly connected graphs, proving a quadratic connectivity sufficiency bound and introducing a powerful packing theorem for rigid spanning subgraphs. Central to the approach is the -dimensional rigidity matroid and its unions, enabling the construction of edge-disjoint -rigid spanning subgraphs under high connectivity. By combining two edge-disjoint minimally -rigid subgraphs with carefully chosen orientations (via a -orientation) and applying Hakimi's orientation criterion, the authors obtain -connected orientations under explicit connectivity. They further adapt these ideas to Kriesell's spanning-tree conjecture, showing that sufficiently connected graphs contain a spanning tree whose removal leaves a -connected graph, with explicit bounds. The results are supported by probabilistic methods, combinatorial matroid theory, and yield algorithmic pathways for constructing the desired packings and orientations.

Abstract

We give an affirmative answer to a long-standing conjecture of Thomassen, stating that every sufficiently highly connected graph has a -vertex-connected orientation. We prove that a connectivity of order suffices. As a key tool, we show that for every pair of positive integers and , every -connected graph contains edge-disjoint -rigid (in particular, -connected) spanning subgraphs, where . This also implies a positive answer to the conjecture of Kriesell that every sufficiently highly connected graph contains a spanning tree such that is -connected.
Paper Structure (9 sections, 20 theorems, 43 equations, 1 figure)

This paper contains 9 sections, 20 theorems, 43 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Rigidity matroids
  4. Unions of rigidity matroids
  5. Tools from probability theory
  6. Packing rigid spanning subgraphs
  7. Highly connected orientations
  8. Removable spanning trees
  9. Concluding remarks

Key Result

Theorem 1.2

Every $(320 \cdot k^2)$-connected graph has a $k$-connected orientation.

Figures (1)

  • Figure 1: The construction of $T$ in \ref{['lemma:graphicrigid']} in the case when $a = 3$ and $d=6$. The complement of $T$ can be obtained from a complete graph $K_{d+1}$ on $d+1$ vertices by successively adding vertices of degree $d$.

Theorems & Definitions (40)

  • Conjecture 1.1
  • Theorem 1.2
  • Conjecture 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Theorem 1.6
  • Theorem 1.7
  • Theorem 1.8
  • Lemma 2.1
  • proof
  • ...and 30 more