A Bijection for Eulerian-equivalence Classes of Totally Cyclic Orientations

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 Eulerianequivalence 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. Key words. Tutte polynomials, reduced orientations, totally cyclic orientations, cycle reversing classes, Eulerian-equivalence classes, internal activity, external activity AMS classification: 05A99, 05C20

• Type: Journal
• Year: 2008
• Where: GC
• Authors: Beifang Chen, Arthur L. B. Yang, Terence Y. J. Zhang
• Keywords: Applied Computing, Corresponding Tutte Polynomial, Cyclic Orientations, GC 2008, Tutte Polynomials