Skip to main content
SpringerLink
Log in

Up-embeddability via girth and the degree-sum of adjacent vertices

  • Published:
Science in China Series A: Mathematics Aims and scope Submit manuscript

Abstract

Let G be a simple graph of order n and girth g. For any two adjacent vertices u and v of G, if d G (u) + d G (v) ⩾ n − 2g + 5 then G is up-embeddable. In the case of 2-edge-connected (resp. 3-edge-connected) graph, G is up-embeddable if d G (u) + d G (v) ⩾ n − 2g + 3 (resp. d G (u) + d G (v) ⩾ n − 2g −5) for any two adjacent vertices u and v of G. Furthermore, the above three lower bounds are all shown to be tight.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Nordhause E A, Stewart B M, White A T. On the maximum genus of a graph. J Combin Theory, 11: 258–267 (1971)

    Google Scholar 

  2. Ringeisen R. The maximum genus of a graph. PhD Thesis. Michigan State University, 1970

  3. Ringeisen R. Upper and lower embeddable graphs. Graph Theory and Applications. In: Lecture Notes in Math. Berlin and New York: Springer-Verlag, 1972, 303

    Google Scholar 

  4. Ringeisen R. Determining all compact orientable 2-manifolds upon which K m,n has 2-cell imbeddings. J Combin Theory Ser B, 12: 101–104 (1974)

    Article  MathSciNet  Google Scholar 

  5. Kundu S. Bounds on number of disjoint spanning trees. J Combin Theory Ser B, 17: 199–203 (1974)

    Article  MATH  MathSciNet  Google Scholar 

  6. Jaeger F, Payan C, Xuong N H. A classes of upper embeddable graphs. J Graph Theory, 3: 387–391 (1979)

    Article  MATH  MathSciNet  Google Scholar 

  7. Jungerment M. A characterization of upper embeddable graphs. Trans Amer Math Soc, 241: 401–406 (1979)

    Article  Google Scholar 

  8. Škoviera M. The maximum genus of graphs diameter two. Discrete Math, 87: 175–180 (1991)

    Article  MATH  MathSciNet  Google Scholar 

  9. Huang Y, Liu Y. Face size and the maximum genus of a graph. J Combin Theory Ser B, 80: 356–370 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  10. Huang Y, Liu Y. The degree sum of nonadjacent vertices and up-embeddability of graphs. Chinese Ann Math Ser A, 5(19): 651–656 (1998)

    Google Scholar 

  11. Bondy J A, Murty U S R. Graph Theory with Applications. London: MacMillan, 1976

    Google Scholar 

  12. Liu Y. Embeddability in Graphs. Dordrecht-Boston-London: Kluwer Academic, 1995, 255–265

    MATH  Google Scholar 

  13. Gross J L, Tucker T W. Topological Graph Theory. New York: Wiley-Interscience, 1987

    MATH  Google Scholar 

  14. White A T. Graphs, Groups, and Surfaces. Amsterdam: North-Holland, 1984

    MATH  Google Scholar 

  15. Liu Y. The maximum orientable genus of a graph. Scientia Sinica (Special Issue), (II): 41–55 (1979)

  16. Xuong N H. How to determine the maximum genus of a graph. J Combin Theory Ser B, 26: 217–225 (1979)

    Article  MATH  MathSciNet  Google Scholar 

  17. Nebeský L. A new characterization of the maximum genus of a graph. Czechoslova Math J, 31(106): 604–613 (1981)

    Google Scholar 

  18. Huang Y. Maximum genus and chromatic number of graphs. Discrete Math, 271: 117–127 (2003)

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

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

    GuangHua Dong & YanPei Liu

  2. Department of Mathematics, School of Science, Tianjin Polytechnic University, Tianjin, 300160, China

    GuangHua Dong

Authors
  1. GuangHua Dong
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. YanPei Liu
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to GuangHua Dong.

Additional information

This work was supported by National Natural Science Foundation of China (Grant No. 10571013)

Rights and permissions

Reprints and permissions

About this article

Cite this article

Dong, G., Liu, Y. Up-embeddability via girth and the degree-sum of adjacent vertices. Sci. China Ser. A-Math. 52, 597–604 (2009). https://doi.org/10.1007/s11425-008-0126-8

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11425-008-0126-8

Keywords

MSC(2000)

Search

Navigation