Journal of Mathematical Sciences

, Volume 212, Issue 6, pp 698–707 | Cite as

On the Structure of C 3-Critical Minimal 6-Connected Graphs

  • A. V. Pastor

In this paper, C 3 -critical minimal 6-connected graphs are studied; they are defined as 6-connected graphs, any subgraph of which obtained by edge deletion is not 6-connected and in which any clique on at most 3 vertices is contained in a 6-cutset. It is proved that more than \( \frac{5}{9} \) of all vertices of such a graph have degree 6. Bibliography: 18 titles.


Standard Notation Adjacent Vertex Critical Graph Pendant Vertex Edge Deletion 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    K. Ando, S. Fujita, and K. Kawarabayashi, “Minimally contraction-critically 6-connected graphs,” Discrete Math., 312, No. 3, 671–679 (2012).zbMATHMathSciNetCrossRefGoogle Scholar
  2. 2.
    K. Ando and T. Iwase, “The number of vertices of degree 5 in a contraction-critically 5-connected graph,” Discrete Math., 311, 1925–1939 (2011).zbMATHMathSciNetCrossRefGoogle Scholar
  3. 3.
    K. Ando, A. Kaneko, and K. Kawarabayashi, “Vertices of degree 6 in a contraction critically 6-connected graphs,” Discrete Math., 273, 55–69 (2003).zbMATHMathSciNetCrossRefGoogle Scholar
  4. 4.
    K. Ando and C. Qin, “Some structural properties of minimally contraction-critically 5-connected graphs,” Discrete Math., 311, 1084–1097 (2011).zbMATHMathSciNetCrossRefGoogle Scholar
  5. 5.
    G. Chartrand, A. Kaugars, and D. R. Lick, “Critically n-connected graphs,” Proc. Amer. Math. Soc., 32, 63–68 (1972).MathSciNetGoogle Scholar
  6. 6.
    M. Fontet, “Graphes 4-essentiels,” C. R. Acad. Se. Paris, 287, Serie A, 289–290 (1978).Google Scholar
  7. 7.
    R. Halin, “A theorem on n-connected graphs,” J. Combin. Theory, 7, 150–154 (1969).zbMATHMathSciNetCrossRefGoogle Scholar
  8. 8.
    M. Li, X. Yuan, and J. Su, “The number of vertices of degree 7 in a contraction-critical 7-connected graph,” Discrete Math., 308, 6262–6268 (2008).zbMATHMathSciNetCrossRefGoogle Scholar
  9. 9.
    W. Mader, “Ecken Vom Gard n in minimalen n-fach zusammenhangenden Graphen,” Arch. Math., 23, 219–224 (1972).zbMATHMathSciNetCrossRefGoogle Scholar
  10. 10.
    W. Mader, “Zur Struktur minimal n-fach zusammenh¨angender Graphen,” Abh. Math. Sem. Univ. Hamburg, 49, 49–69 (1979).zbMATHMathSciNetCrossRefGoogle Scholar
  11. 11.
    W. Mader, “Generalization of critical connectivity of graphs,” Discrete Math., 72, 267–283 (1988).zbMATHMathSciNetCrossRefGoogle Scholar
  12. 12.
    N. Martinov, “A recursive characterization of the 4-connected graphs,” Discrete Math., 84, 105–108 (1990).zbMATHMathSciNetCrossRefGoogle Scholar
  13. 13.
    W. T. Tutte, “A theory of 3-connected graphs,” Indag. Math., 23, 441–455 (1961).MathSciNetCrossRefGoogle Scholar
  14. 14.
    D. V. Karpov and A. V. Pastor, “On the structure of a k-connected graph,” Zap. Nauchn. Semin. POMI, 266, 76-106 (2000).zbMATHGoogle Scholar
  15. 15.
    S. A. Obraztsova, “Local structure of 5 and 6-connected graphs,” Zap. Nauchn. Semin. POMI, 381, 88-96 (2010).Google Scholar
  16. 16.
    S. A. Obraztsova and A. V. Pastor, “Local structure of 7 and 8-connected graphs,” Zap. Nauchn. Semin. POMI, 381, 97–111 (2010).Google Scholar
  17. 17.
    S. A. Obraztsova, “Local structure of 9 and 10-connected graphs,” Zap. Nauchn. Semin. POMI, 391, 157–197 (2011).Google Scholar
  18. 18.
    S. A.Obraztsova and A.V. Pastor, “About vertices of degree k of minimally and contraction critically k-connected graphs: upper bounds,” Zap. Nauchn. Semin. POMI, 391, 198–210 (2011).Google Scholar

Copyright information

© Springer Science+Business Media New York 2016

Authors and Affiliations

  1. 1.St. Petersburg Department of the Steklov Mathematical InstituteSt. Petersburg State Polytechnic UniversitySt. PetersburgRussia

Personalised recommendations