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Super Connectivity of the Generalized Mycielskian of Graphs

  • S. Francis Raj
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7643)

Introduction

All graphs considered in this paper are simple, finite, nontrivial and undirected.

Let G be a graph with vertex set \(V^0=\{v_0^0,v_1^0,\ldots,v_{n-1}^0\}\) and edge set E 0. Given an integer m ≥ 1, the m-Mycielskian (also known as the generalized Mycielskian) of G, denoted by μ m (G), is the graph whose vertex set is the disjoint union

$$V^0\cup V^1\cup\ldots\cup V^m\cup\{u\},$$
where \(V^i=\{v_j^i;v_j^0\in V^0\}\) is the i-th copy of V 0, i = 1,2,…,m, and edge set
$$E^0\cup\Big(\mathop{\cup}\limits_{i=0}^{m-1}\{v_j^iv_{j'} ^{i+1}\ :v_j^0v_{j'}^0\in E^0\}\Big)\cup \{v_j^mu: v_j^m\in V^m\}.$$

Keywords

Mycielskian Generalized Mycielskian Vertex-connectivity Edge- connectivity Super connectivity Super edge connectivity 

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References

  1. 1.
    Balakrishanan, R., Francis Raj, S.: Connectivity of the Mycielskian of a graph. Discrete Math. 308, 2607–2610 (2007)CrossRefGoogle Scholar
  2. 2.
    Balakrishnan, R., Ranganathan, K.: A Textbook of Graph Theory. Springer, New York (2000)zbMATHCrossRefGoogle Scholar
  3. 3.
    Fisher, D.C., McKena, P.A., Boyer, E.D.: Hamiltonicity, diameter, domination, packing and biclique partitions of the Mycielski’s graphs. Discrete Appl. Math. 84, 93–105 (1998)MathSciNetzbMATHCrossRefGoogle Scholar
  4. 4.
    Francis Raj, S.: Connectivity of the generalised Mycielskian of digraphs, Graphs and Combin., doi: 10.1007/s00373-012-1151-5Google Scholar
  5. 5.
    Guo, L., Liu, R., Guo, X.: Super Connectivity and Super Edge Connectivity of the Mycielskian of a Graph. Graphs and Combin. 28, 143–147 (2012)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Liu, J., Meng, J.: Super-connected and super-arc-connected Cartesian product of digraphs. Inform. Process. Lett. 108, 90–93 (2008)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Lam, P.C.B., Gu, G., Lin, W., Song, Z.: Circular Chromatic Number and a generalization of the construction of Mycielski. J. Combin. Theory, Ser. B 89, 195–205 (2003)MathSciNetzbMATHCrossRefGoogle Scholar
  8. 8.
    Lin, W., Wu, J., Lam, P.C.B., Gu, G.: Several parameters of generalised Mycielskians. Discrete Appl. Math. 154, 1173–1182 (2006)MathSciNetzbMATHCrossRefGoogle Scholar
  9. 9.
    Mycielski, J.: Sur le colouriage des graphes. Colloq. Math. 3, 161–162 (1955)MathSciNetzbMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • S. Francis Raj
    • 1
  1. 1.Department of MathematicsPondicherry UniversityPuducherryIndia

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