Embeddings of Pfaffian Braces and Polyhex Graphs

Details Embeddings of Pfaffian Braces and Polyhex Graphs Embeddings of Pfaffian Braces and Polyhex Graphs

2009-08-11

arxiv.org/abs/0908.1506v1

Mathematics

Combinatorics

17 pages, 13 figures

Scientific paper

Let $G$ be a graph admitting a perfect matching. A cycle of even size $C$ is central if $G-C$ has a perfect matching. Given an orientation to $G$, an even cycle $C$ is oddly oriented if along either direction of traversal around $C$, the number of edges of $C$ with the direction as the same as the traversal direction is odd. An orientation of $G$ is Pfaffian if every central cycle of $G$ is oddly oriented. A graph $G$ is Pfaffian if it has a Pfaffian orientation. In this paper, we show that every embedding of a Pfaffian brace on a surface with positive genus has face-width at most three and that the cyclic edge-connectivity of a Pfaffian cubic brace different from the Heawood graph is four. Finally, we characterize all Pfaffian polyhex graphs.

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