Journal of Combinatorial Optimization

, Volume 27, Issue 3, pp 530–540 | Cite as

The Pfaffian property of Cartesian products of graphs



Suppose that G=(V,E) is a graph with even vertices. An even cycle C is a nice cycle of G if GV(C) has a perfect matching. An orientation of G is a Pfaffian orientation if each nice cycle C has an odd number of edges directed in either direction of the cycle. Let P n and C n denote the path and the cycle on n vertices, respectively. In this paper, we characterize the Pfaffian property of Cartesian products G×P 2n and G×C 2n for any graph G in terms of forbidden subgraphs of G. This extends the results in (Yan and Zhang in Discrete Appl Math 154:145–157, 2006).


Perfect matchings Pfaffian graphs Cartesian products of graphs 


  1. Fischer I, Little CHC (2001) A characterisation of Pfaffian near bipartite graphs. J Comb Theory, Ser B 82:175–222 CrossRefMATHMathSciNetGoogle Scholar
  2. Fischer I, Little CHC (2003) Even circuits of prescribed clockwise parity. Electron J Comb 10:#R45 MathSciNetGoogle Scholar
  3. Kasteleyn PW (1961) The statistics of dimers on a lattice. Physica A 12:1209–1225 Google Scholar
  4. Kasteleyn PW (1967) Graph theory and crystal physics. In: Harary F (ed) Graph theory and theoretical physics. Academic Press, San Diego, pp 43–110 Google Scholar
  5. Lin F, Zhang L (2009) Pfaffian orientation and enumeration of perfect matchings for some Cartesian products of graphs. Electron J Comb 16:#R52 Google Scholar
  6. Little CHC (1975) A characterization of convertible (0,1)-matrices. J Comb Theory, Ser B 18:187–208 CrossRefMATHMathSciNetGoogle Scholar
  7. Little CHC, Rendl F (1991) Operations preserving the Pfaffian property of a graph. J Aust Math Soc A 50:248–257 CrossRefMATHMathSciNetGoogle Scholar
  8. Lovász L, Plummer M (1986) Matching theory. Ann of Discrete Math, vol 29. North-Holland, New York MATHGoogle Scholar
  9. McCuaig W, Robertson N, Seymour PD, Thomas, R (1997) Permanents, Pfaffian orientations, and even directed circuits (Extended abstract). In: Proc symposium on the theory of computing (STOC) Google Scholar
  10. Norine S, Thomas R (2008) Minimally non-Pfaffian graphs. J Comb Theory, Ser B 98:1038–1055 CrossRefMATHMathSciNetGoogle Scholar
  11. Robertson R, Seymour PD, Thomas R (1999) Permanents, Pfaffian orientations, and even directed circuits. Ann Math 150:929–975 CrossRefMATHMathSciNetGoogle Scholar
  12. Yan W, Zhang F (2004) Enumeration of perfect matchings of graphs with reflective symmetry by Pfaffians. Adv Appl Math 32:655–668 CrossRefMATHGoogle Scholar
  13. Yan W, Zhang F (2006) Enumeration of perfect matchings of a type of Cartesian products of graphs. Discrete Appl Math 154:145–157 CrossRefMATHMathSciNetGoogle Scholar
  14. Zhang L, Wang Y, Lu F (2012) Pfaffian graphs embedding on the torus (submitted) Google Scholar

Copyright information

© Springer Science+Business Media, LLC 2012

Authors and Affiliations

  1. 1.School of SciencesLinyi UniversityLinyiChina
  2. 2.School of Mathematical SciencesXiamen UniversityXiamenChina

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