Abstract
Gioan showed that the number of cycle reversing classes of totally cyclic orientations of a given graph can be calculated as an evaluation of the corresponding Tutte polynomial. We note that the concept of cycle reversing classes of orientations coincides with that of Eulerian-equivalence classes considered by Chen and Stanley, and Kochol. Based on this coincidence, we give a bijective proof of Gioan’s result. Precisely, the main result of the paper is an algorithmic bijection between the set of Eulerian-equivalence classes of totally cyclic orientations and the set of spanning trees without internally active edges.
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Chen, B., Yang, A.L.B. & Zhang, T.Y.J. A Bijection for Eulerian-equivalence Classes of Totally Cyclic Orientations. Graphs and Combinatorics 24, 519–530 (2008). https://doi.org/10.1007/s00373-008-0813-9
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DOI: https://doi.org/10.1007/s00373-008-0813-9
Keywords
- Tutte polynomials
- reduced orientations
- totally cyclic orientations
- cycle reversing classes
- Eulerian-equivalence classes
- internal activity
- external activity