# The Pfaffian property of graphs on the Möbius strip based on topological resolution

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## Abstract

Recently, the problem about Pfaffian property of graphs has attracted much attention in the matching theory. Its significance stems from the fact that the number of perfect matchings in a graph *G* can be computed in polynomial time if *G* is Pfaffian. The Möbius strip with unique surface structure strip has potentially significant chirality properties in molecular chemistry. In this paper, we consider the Pfaffian property of graphs on the Möbius strip based on topological resolution. A sufficient condition for Pfaffian graphs on the Möbius strip is obtained. As its application, we characterize Pfaffian quadrilateral lattices on the Möbius strip.

## Keywords

Möbius strip Pfaffian property Drawing in the plane with crossings## Mathematics Subject Classification

05C10 05C70 05C75 92D20## Notes

### Acknowledgements

This work is partially supported by NSF of China(No. 11671186), NSF of Fujian Province (2017J01404) and Science Foundation for the Education Department of Fujian Province (JZ160455). It is also supported by the Institute of Meteorological Big Data-Digital Fujian and Fujian Key Laboratory of Data Science and Statistics.

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