Kastelyn's Theorem and Noneven Symmetric Digraphs
نویسنده
چکیده
A noneven digraph is a directed graph with self-loops such that at least one weight-ing of its arcs by 0 or 1 results in no even cycles. A directed cycle in a weighted digraph is even if the sum of weights of its arcs is an even number. The characterisation of 2-connected undirected graphs G such that their symmetric directed counterparts G are noneven, is obtained via a result of Kastelyn concerning Pfaaan orientations of graphs. An undirected graph has Pfaaan orientation if its edges could be oriented in such a way that each of its cycle has an odd number of edges that are oriented along the direction of traverse. Kastelyn proved that all planar graphs have Pfaaan orientation, and used his result to give a combinatorial dimer solution of the 2-D Ising model.
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