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Graphical condensation of plane graphs: a combinatorial approach

Weigen Yan·Yeong-Nan Yeh·Fuji Zhang
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Abstract

The method of graphical vertex-condensation for enumerating perfect matchings of plane bipartite graph was found by Propp (Theoret. Comput. Sci. 303(2003), 267-301), and was generalized by Kuo (Theoret. Comput. Sci. 319 (2004), 29-57) and Yan and Zhang (J. Combin. Theory Ser. A, 110(2005), 113-125). In this paper, by a purely combinatorial method some explicit identities on graphical vertex-condensation for enumerating perfect matchings of plane graphs (which do not need to be bipartite) are obtained. As applications of our results, some results on graphical edge-condensation for enumerating perfect matchings are proved, and we count the sum of weights of perfect matchings of weighted Aztec diamond.

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Weigen Yan, Yeong-Nan Yeh, Fuji Zhang. 2005-09-23. Graphical condensation of plane graphs: a combinatorial approach. https://arxiv.org/abs/math/0509337

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