Combinatorica

, Volume 33, Issue 2, pp 231–252 | Cite as

Bipartite partial duals and circuits in medial graphs

Original Paper

Abstract

It is well known that a plane graph is Eulerian if and only if its geometric dual is bipartite. We extend this result to partial duals of plane graphs. We then characterize all bipartite partial duals of a plane graph in terms of oriented circuits in its medial graph.

Mathematics Subject Classification (2000)

05C10 05C45 

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References

  1. [1]
    A. Asratian, T. Denley and R. Häggkvist: Bipartite graphs and their applications, Cambridge Tracts in Mathematics, 131. Cambridge University Press, Cambridge, 1998.MATHCrossRefGoogle Scholar
  2. [2]
    J. Bondy and U. Murty: Graph theory, Graduate Texts in Mathematics 244, Springer, Berlin, 2008.Google Scholar
  3. [3]
    S. Chmutov: Generalized duality for graphs on surfaces and the signed Bollobás-Riordan polynomial, J. Combin. Theory Ser. B 99 (2009) 617–638 arXiv:0711.3490.MathSciNetMATHCrossRefGoogle Scholar
  4. [4]
    J. Edmonds: On the surface duality of linear graphs, J. Res. Nat. Bur. Standards Sect. B 69B (1965) 121–123.MathSciNetCrossRefGoogle Scholar
  5. [5]
    J. A. Ellis-Monaghan and I. Moffatt: Twisted duality for embedded graphs, Trans. Amer. Math. Soc. 364 (2012) 1529–1569.MathSciNetMATHCrossRefGoogle Scholar
  6. [6]
    S. Huggett, I. Moffatt and N. Virdee: On the graphs of link diagrams and their parallels, Math. Proc. Cambridge Philos. Soc., 153 (2012) 123–145 arXiv:1106.4197.MathSciNetMATHCrossRefGoogle Scholar
  7. [7]
    F. Jaeger: A note on sub-Eulerian graphs, J. Graph Theory 3 (1979) 91–93.MathSciNetMATHCrossRefGoogle Scholar
  8. [8]
    T. Krajewski, V. Rivasseau and F. Vignes-Tourneret: Topological graph polynomials and quantum field theory, Part II: Mehler kernel theories, ann. Henri Poincaré 12 (2011) 1–63 arXiv:0912.5438.Google Scholar
  9. [9]
    I. Moffatt: Unsigned state models for the Jones polynomial, Ann. Comb. 15 (2011) 127–146 arXiv:0710.4152.MathSciNetMATHCrossRefGoogle Scholar
  10. [10]
    I. Moffatt: Partial duality and Bollobás and Riordan’s ribbon graph polynomial, Discrete Math. 310 (2010) 174–183 arXiv:0809.3014.MathSciNetMATHCrossRefGoogle Scholar
  11. [11]
    I. Moffatt: A characterization of partially dual graphs, J. Graph Theory 67 (2011) 198–217 arXiv:0901.1868.MathSciNetMATHCrossRefGoogle Scholar
  12. [12]
    I. Moffatt: Partial duals of plane graphs, separability and the graphs of knots, Algebr. Geom. Topol. 12 (2012) 1099–1136 arXiv:1007.4219.MathSciNetMATHCrossRefGoogle Scholar
  13. [13]
    J. van Lint and R. Wilson: A course in combinatorics, Cambridge University Press, Cambridge, 2001.MATHCrossRefGoogle Scholar
  14. [14]
    F. Vignes-Tourneret: The multivariate signed Bollobás-Riordan polynomial, Discrete Math. 309 (2009) 5968–5981 arXiv:0811.1584.MathSciNetMATHCrossRefGoogle Scholar
  15. [15]
    F. Vignes-Tourneret: Non-orientable quasi-trees for the Bollobás-Riordan polynomial, European J. Combin. 32 (2011) 510–532.MathSciNetMATHCrossRefGoogle Scholar
  16. [16]
    D. Welsh: Euler and bipartite matroids, J. Combinatorial Theory 6 (1969) 375–377.MathSciNetMATHCrossRefGoogle Scholar
  17. [17]
    H. Whitney: Non-separable and planar graphs, Trans. Amer. Math. Soc. 34 (1932) 339–362.MathSciNetCrossRefGoogle Scholar

Copyright information

© János Bolyai Mathematical Society and Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  1. 1.School of Computing and MathematicsUniversity of PlymouthDevonUK
  2. 2.Department of Mathematics Royal HollowayUniversity of LondonEgham, SurreyUK

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