A counterexample to a result on the tree graph of a graph

TL;DR

This paper presents an infinite family of counterexamples to a previously claimed theorem about the connectivity of tree graphs defined by cycle families in 2-connected graphs.

Contribution

It provides counterexamples to Hu's theorem, challenging the assumption that certain cycle families guarantee connectivity of the tree graph.

Findings

Counterexamples invalidate Hu's theorem.

Connectivity of T(G,C) does not always follow from the cycle family properties.

The result clarifies limitations in the theory of cycle spaces and tree graphs.

Abstract

Given a set of cycles C of a graph G, the tree graph of G defined by C is the graph T(G,C) whose vertices are the spanning trees of G and in which two trees R and S are adjacent if the union of R and S contains exactly one cycle and this cycle lies in C. Li et al [Discrete Math 271 (2003), 303--310] proved that if the graph T(G,C) is connected, then C cyclically spans the cycle space of G. Later, Yumei Hu [Proceedings of the 6th International Conference on Wireless Communications Networking and Mobile Computing (2010), 1--3] proved that if C is an arboreal family of cycles of G which cyclically spans the cycle space of a $2$-connected graph G, then T(G, C) is connected. In this note we present an infinite family of counterexamples to Hu's result.