Dualities and reciprocities on graphs on surfaces |
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| Institution: | 1. Department of Mathematics, Sogang University, Seoul, South Korea;2. Department of Mathematics, Seoul National University, Seoul, South Korea |
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| Abstract: | We extend the duality between acyclic orientations and totally cyclic orientations on planar graphs to dualities on graphs on orientable surfaces by introducing boundary acyclic orientations and totally bi-walkable orientations. In addition, we provide a reciprocity theorem connecting local tensions and boundary acyclic orientations. Furthermore, we define the balanced flow polynomial which is connected with tension polynomial by duality and with totally bi-walkable orientations by reciprocity. |
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| Keywords: | Graph on a surface Tension Flow Acyclic orientation Totally cyclic orientation Duality Ehrhart theory Combinatorial reciprocity |
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