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arXiv · 1511.01372

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

Ana Paulina Figueroa·Eduardo Rivera-Campo
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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.

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Ana Paulina Figueroa, Eduardo Rivera-Campo. 2015-11-04. A counterexample to a result on the tree graph of a graph. https://arxiv.org/abs/1511.01372

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