Face-Width of Pfaffian Braces and Polyhex Graphs on Surfaces

A graph G with a perfect matching is Pfaffian if it admits an orientation D such that every central cycle C (i.e. C is of even size and G − V (C) has a perfect matching) has an odd number of edges oriented in either direction of the cycle. It is known that the number of perfect matchings of a Pfaffian graph can be computed in polynomial time. In this paper, we show that every embedding of a Pfaffian brace (i.e. 2-extendable bipartite graph) on a surface with a positive genus has face-width at most 3. Further, we study Pfaffian cubic braces and obtain a characterization of Pfaffian polyhex graphs: a polyhex graph is Pfaffian if and only if it is either non-bipartite or isomorphic to the cube, or the Heawood graph, or the Cartesian product Ck ×K2 for even integers k > 6.