Non-contractible edges in a 3-connected graph

Abstract

An edgee in a 3-connected graphG is contractible if the contraction ofe inG results in a 3-connected graph; otherwisee is non-contractible. In this paper, we prove that the number of non-contractible edges in a 3-connected graph of orderp≥5 is at most

$$3p - \left[ {\frac{3}{2}(\sqrt {24p + 25} - 5} \right],$$

and show that this upper bound is the best possible for infinitely many values ofp.

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References

  1. [1]

    R. E. L. Aldred, R. L. Hemminger, andX. Yu: The 3-connected graphs with a maximum matching containing precisely one contractible edge,J. Graph Theory, in press.

  2. [2]

    K. Ando, H. Enomoto, andA. Saito: Contractible edges in 3-connected graphs,J. Comb. Theory Ser. B 42 (1987), 87–93.

    Google Scholar 

  3. [3]

    J. A. Bondy, andU. S. R. Murty:Graph Theory with Applications, American Elsevier (1976).

  4. [4]

    N. Dean: Contractible edges and conjectures about path and cycle numbers,Ph.D Thesis, Vanderbilt University (1987).

  5. [5]

    R. L. Hemminger, andX. Yu: 3-Connected graphs with contractible edge covers of sizek, Discrete Math. 101 (1992), 115–133.

    Google Scholar 

  6. [6]

    T. B. Kirkman: On a problem in combinations,Cambridge and Dublin Math. J. 2 (1847), 191–204.

    Google Scholar 

  7. [7]

    M. D. Plummer, andB. Toft: Cyclic colorations of 3-polytopes,J. Graph Theory 11 (1987), 507–515.

    Google Scholar 

  8. [8]

    C. Thomassen: Planarity and duality of finite and infinite graphs,J. Comb. Theory Ser. B 29 (1980), 244–271.

    Google Scholar 

  9. [9]

    W. T. Tutte: A theory of 3-connected graphs,Indag. Math. 23 (1961), 441–455.

    Google Scholar 

  10. [10]

    X. Yu: 3-Connected graphs with non-cut contractible edge covers of sizek, J. Graph Theory, in press.

  11. [11]

    X. Yu: Non-separating cycles and discrete Jordan curves,J. Comb. Theory Ser. B 54 (1992), 142–154.

    Google Scholar 

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Partially supported by Nihon University Research Grant B90-026

Partially supported by NSF under grant No. DMS-9105173

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Egawa, Y., Ota, K., Saito, A. et al. Non-contractible edges in a 3-connected graph. Combinatorica 15, 357–364 (1995). https://doi.org/10.1007/BF01299741

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Mathemacics Subject Classification (1991)

  • 05 C 40