DMGT

ISSN 1234-3099 (print version)

ISSN 2083-5892 (electronic version)

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Discussiones Mathematicae Graph Theory

ON THE TREE GRAPH OF A CONNECTED GRAPH

 Ana Paulina Figueroa Instituto de Matemáticas Universidad Nacional Autónoma de México Ciudad Universitaria, México D.F. 04510, México Eduardo Rivera-Campo Departmento de Matemáticas Universidad Autónoma Metropolitana-Iztapalapa Av. San Rafael Atlixco 186, México D.F. 09340, México

Abstract

Let G be a graph and C be a set of cycles of G. The tree graph of G defined by C, is the graph T(G,C) that has one vertex for each spanning tree of G, in which two trees T and T′ are adjacent if their symmetric difference consists of two edges and the unique cycle contained in T∪T′ is an element of C. We give a necessary and sufficient condition for this graph to be connected for the case where every edge of G belongs to at most two cycles in C.

Keywords: tree graph, property Δ*, property Δ+.

2000 Mathematics Subject Classification: 05C05.

References

 [1] H.J. Broersma and X. Li, The connectivity of the of the leaf-exchange spanning tree graph of a graph, Ars. Combin. 43 (1996) 225-231. [2] F. Harary, R.J. Mokken and M. Plantholt, Interpolation theorem for diameters of spanning trees, IEEE Trans. Circuits and Systems 30 (1983) 429-432, doi: 10.1109/TCS.1983.1085385. [3] K. Heinrich and G. Liu, A lower bound on the number of spanning trees with k endvertices, J. Graph Theory 12 (1988) 95-100, doi: 10.1002/jgt.3190120110. [4] X. Li, V. Neumann-Lara and E. Rivera-Campo, On a tree graph defined by a set of cycles, Discrete Math. 271 (2003) 303-310, doi: 10.1016/S0012-365X(03)00132-8. [5] F.J. Zhang and Z. Chen, Connectivity of (adjacency) tree graphs, J. Xinjiang Univ. Natur. Sci. 3 (1986) 1-5.