Skip to main content
SpringerLink
Log in

Connectivity of the Generalised Mycielskian of Digraphs

  • Original Paper
  • Published:
Graphs and Combinatorics Aims and scope Submit manuscript

Abstract

In a search for triangle-free graphs with arbitrarily large chromatic number, Mycielski developed a graph transformation that transforms a graph G into a new graph μ(G), which is called the Mycielskian of G. A generalisation of this transformation is the generalised Mycielskian μ m (G), m a positive integer. This paper investigates the vertex-connectivity κ and arc-connectivity κ′ of the generalised Mycielskian of strongly connected digraphs D. We show that κ (μ m (D)) = min{δ(μ m (D)), (m + 1)κ (D) + 1} and κ′ (μ m (D)) = δ(μ m (D)) where δ(μ m (D)) denotes the minimum degree of the generalised Mycielisian μ m (D). This turns out to be a generalisation of the results due to Guo and Guo (Appl. Math. Lett. 22:1622–1625, 2009).

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. Balakrishanan R., Francis Raj S.: Connectivity of the Mycielskian of a graph. Discret. Math. 308, 2607–2610 (2007)

    Article  Google Scholar 

  2. Balakrishnan R., Ranganathan K.: A Textbook of Graph Theory. Springer, New York (2000)

    Book  MATH  Google Scholar 

  3. Caramia M., Dell’Olmo P.: A lower bound on the chromatic number of Mycielski graphs. Discret. Math 235, 79–86 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  4. Chang G.J., Huang L., Zhu X.: Circular chromatic number of Mycielski's graphs. Discret. Math 205, 23–37 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  5. Cropper M., Gyárfás A., Lehel J.: Hall ratio of theMycielski graphs. Discret. Math. 306, 1988–1990 (2006)

    Article  MATH  Google Scholar 

  6. Fisher D.C., McKena P.A., Boyer E.D.: Hamiltonicity, diameter, domination, packing and biclique partitions of the Mycielski’s graphs. Discret. Appl. Math. 84, 93–105 (1998)

    Article  MATH  Google Scholar 

  7. Guo L., Guo X.: Connectivity of the Mycielskian of a digraph. Appl. Math. Lett 22, 1622–1625 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  8. Lam, P.C.B.,Gu, G., Lin,W., Song, Z.: Some properties of generalized Mycielski’s graphs (manuscript)

  9. 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)

    Article  MathSciNet  MATH  Google Scholar 

  10. Larsen M., Propp , J. , Ullman D.: The fractional chromatic number of Mycielski’s graphs. J. Graph Theory 19, 411–416 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  11. Lin W., Wu J., Lam P.C.B., Gu G.: Several parameters of generalised Mycielskians. Discret. Appl. Math. 154, 1173–1182 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  12. Mycielski J.: Sur le colouriage des graphes. Colloq. Math 3, 161–162 (1955)

    MathSciNet  MATH  Google Scholar 

Download references

Author information

Author notes

  1. S. Francis Raj

    Present address: Department of Mathematics, Pondicherry University, Puducherry, 605014, India

Authors and Affiliations

  1. Department of Mathematics, Bharathidasan University, Tiruchirappalli, 620024, India

    S. Francis Raj

Authors
  1. S. Francis Raj
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to S. Francis Raj.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Francis Raj, S. Connectivity of the Generalised Mycielskian of Digraphs. Graphs and Combinatorics 29, 893–900 (2013). https://doi.org/10.1007/s00373-012-1151-5

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00373-012-1151-5

Keywords

Mathematics Subject Classification

Search

Navigation