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The genus polynomials of cross-ladder digraphs in orientable surfaces

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Abstract

Some results about the genus distributions of graphs are known, but little is known about those of digraphs. In this paper, the method of joint trees initiated by Liu is generalized to compute the embedding genus distributions of digraphs in orientable surfaces. The genus polynomials for a new kind of 4-regular digraphs called the cross-ladders in orientable surfaces are obtained. These results are close to solving the third problem given by Bonnington et al.

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References

  1. Archdeacon D. Topological graph theory, a survey. Congr Numer, 115: 5–54 (1996)

    MATH  MathSciNet  Google Scholar 

  2. Xoung N H. How to determine the maximum genus of graph. J Combin Theory, 26: 217–225 (1979)

    Article  Google Scholar 

  3. Nebesky L. Characterizing the maximal genus of connected graphs. Czechoslovak Math J, 43: 177–185 (1993)

    MATH  MathSciNet  Google Scholar 

  4. Liu Y P. Embeddability in Graphs. Dordrecht-Boston-London: Kluwer, 1995

    MATH  Google Scholar 

  5. Huang Y Q, Liu Y P. Statman the maximum genus of graphs with diameter three. Discrete Math, 194: 139–149 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  6. Furst M L, Gross G L, McGeoch L A. Finding a maximum genus graph imbedding. J ACM, 35: 523–534 (1988)

    Article  MathSciNet  Google Scholar 

  7. Thomassen C. The graph genus problem is NP-complete. J Algorithms, 10: 568–576 (1989)

    Article  MATH  MathSciNet  Google Scholar 

  8. Gross G L, Furst M L. Hierarcy of imbedding distribution invariants of a graph. J Graph theory, 11: 205–220 (1987)

    Article  MATH  MathSciNet  Google Scholar 

  9. McGeoch L A. Algorithms for two graph problems: computing maximum genus imbedding and the two-server problem. PhD Thesis. Pittsburgh, PA: Computer Science Dept, Carnegie Mellon University, 1987

    Google Scholar 

  10. Furst M L, Gross G L, Statman R. Genus distributions for two classes of graphs. J Combin Theory, 46: 22–36 (1989)

    Article  MATH  MathSciNet  Google Scholar 

  11. Gross J L, Robbins D P, Tucker T W. Genus distributions for bouquets of circles. J Combin Theory, 47: 292–306 (1989)

    Article  MATH  MathSciNet  Google Scholar 

  12. Tesar E H. Genus distribution for Ringel ladders. Discrete Math, 216: 235–252 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  13. Chen J E, Gross J L, Rieper R G. Overlap matrices and total imbedding distributions. Discrete Math, 128: 73–94 (1994)

    Article  MATH  MathSciNet  Google Scholar 

  14. Kwak J H, Shim S H. Total embedding distributions for bouquets of circles. Discrete Math, 248: 93–108 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  15. Bonnington C P, Conder M, Morton M. Embedding digraphs on orientable surfaces. J Combin Theory, 85: 1–20 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  16. Bonnington C P, Hartsfield N, Siran J. Obstructions to directed embeddings of Eulerian digraphs in the plane. J Combin Theory, 25: 877–891 (2004)

    MATH  MathSciNet  Google Scholar 

  17. Liu Y P. Advances in Combinatorial Maps (in Chinese). Beijing: Northern Jiaotong University Press, 2003

    Google Scholar 

  18. Hao R X, Liu Y P. The genus distribution of directed antiladders in orientable surfaces. Appl Math Lett, 21: 161–164 (2008)

    Article  MATH  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, Beijing Jiaotong University, Beijing, 100044, China

    RongXia Hao & YanPei Liu

Authors
  1. RongXia Hao
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  2. YanPei Liu
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Corresponding author

Correspondence to RongXia Hao.

Additional information

This work was supported by Beijing Jiaotong University Fund (Grant No. 2004SM054) and the National Natural Science Foundation of China (Grant No. 10571013)

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Hao, R., Liu, Y. The genus polynomials of cross-ladder digraphs in orientable surfaces. Sci. China Ser. A-Math. 51, 889–896 (2008). https://doi.org/10.1007/s11425-007-0125-1

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  • DOI: https://doi.org/10.1007/s11425-007-0125-1

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