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Abstract: Kasteleyn stated that the generating function of the perfect matchings of a graph of genus g may be written as a linear combination of 4 g Pfaffians. Here we prove this statement. As a consequence we present a combinatorial way to compute permanents of square matrices. Other consequences will be presented in a forthcoming paper. Supported by DONET, GACR 0194 and GAUK 194 0 1 Introduction We present a theory of Pfaffian orientations of graphs, introduced by Kasteleyn ([7, 6, 5]). Our... (Update)
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BibTeX entry: (Update)
A. Galluccio and M. Loebl. A theory of pfaffian orientations I: Perfect matchings and permanents. Technical Report 452, IASI-CNR, 1997. http://citeseer.ist.psu.edu/galluccio97theory.html More
@misc{ galluccio97theory,
author = "A. Galluccio and M. Loebl",
title = "A theory of pfaffian orientations I: Perfect matchings and permanents",
text = "A. Galluccio and M. Loebl. A theory of pfaffian orientations I: Perfect
matchings and permanents. Technical Report 452, IASI-CNR, 1997.",
year = "1997",
url = "citeseer.ist.psu.edu/galluccio97theory.html" }
Citations (may not include all citations):251 The complexity of computing the permanent (context) - Valiant - 1979
41 Graph theory and crystal physics (context) - Kasteleyn - 1967
35 Dimer statistics and phase transitions (context) - Kasteleyn - 1963
30 The statistics of dimers on a lattice (context) - Kasteleyn - 1961
23 and even directed circuits (context) - McCuaig, Robertson et al. - 1996
12 Sur les determinants gauches (context) - Cayley
10 A combinatorial solution of the two-dimensional Ising model (context) - Kac, Ward - 1952
9 Balancing signed toroidal graphs in polynomial time (context) - Barahona - 1983
6 permanents and even cycles in directed graphs (context) - Vazirani, Yannakakis - 1989
6 The combinatorial method and the two-dimensional Ising model (context) - Potts, Ward - 1955
5 Lozung zu (context) - Szego - 1913
4 Even cycles and H-homeomorphisms (context) - Galluccio, Loebl - 1995
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