A Diameter-Revealing Proof of the Bondy-Lov\'asz Lemma

TL;DR

This paper provides a new, diameter-revealing proof of a lemma related to the connectivity of a graph of spanning trees, with implications for graph partitioning and an asymptotically tight diameter bound.

Contribution

It offers a strengthened, constructive proof of the Bondy-Lovász lemma, establishing an asymptotically tight diameter bound for the spanning tree graph.

Findings

Constructive proof of the strengthened lemma.

Asymptotically tight O(|V|^2) diameter bound.

Implications for the k=2 case of the Győri-Lovász Theorem.

Abstract

We present a strengthened version of a lemma due to Bondy and Lov\'asz. This lemma establishes the connectivity of a certain graph whose nodes correspond to the spanning trees of a 2-vertex-connected graph, and implies the k=2 case of the Gy\H{o}ri-Lov\'asz Theorem on partitioning of k-vertex-connected graphs. Our strengthened version constructively proves an asymptotically tight O(|V|^2) bound on the worst-case diameter of this graph of spanning trees.