Skip to main content
Log in

Sizes of Critical Graphs with Small Maximum Degrees

  • Published:
Graphs and Combinatorics Aims and scope Submit manuscript

Abstract

In this paper, we give new lower bounds for the size of Δ-critical graphs with Δ=8,9 which improve the partial results of Luo [6] and Y. Zhao [12].

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

Access this article

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

Instant access to the full article PDF.

Similar content being viewed by others

References

  1. Beineke, L.W., Fiorini, S.: On small graphs critical with respect to edge-colourings. Discrete Math. 16, 109–121 (1976)

    Google Scholar 

  2. Clark, L., Haile, D.: Remark on the size of critical edge chromatic graphs. Discrete Math. 171, 287–293 (1997)

    Google Scholar 

  3. Fiorini, S.: Some remarks on a paper by Vizing on critical graphs, Math. Proc. Camb. Phil. Soc. 77, 475–483 (1975)

    Google Scholar 

  4. Jacobsen, I.T.: On critical graphs with chromatic index 4. Discrete Math. 9, 265–276 (1974)

    Google Scholar 

  5. Kayathri, K.: On the size of edge-chromatic critical graphs. Graphs and Combinatorics. 10, 139–144 (1994)

    Google Scholar 

  6. Luo, R., Zhang, C.Q.: Edge coloring of graphs with small average degrees. Discrete Math. 275, 207–218 (2004)

    Google Scholar 

  7. Luo, R.: personal communication

  8. Sanders, D., Zhao, Y.: Planar graphs of maximum degree seven are class I, J. Combin. Theory Ser. B. 83(2), 201–212 (2001)

    Google Scholar 

  9. Sanders, D., Zhao, Y.: On the size of edge chromatic critical graphs, J. Combin. Theory Ser. B. 86, 408–412 (2002)

    Google Scholar 

  10. Vizing, V.G.: On an estimate of the chromatic class of a p-graph. Metody Diskret. Analiz. 3, 25–30 (1964)

    Google Scholar 

  11. Vizing, V.G.: Some unsolved problems in graph theory (in Russian). Uspekhi Mat. Nauk. 23, 117–134 (1968); English translation in Russian Math. Surveys 23, 125–141 (1968)

  12. Zhao, Y.: New Lower Bounds for the size of edge chromatic critical graphs, J. Graphs Theory. 46, 81–92 (2004)

    Google Scholar 

  13. Zhang, L.: Every planar graph with maximum degree 7 is of class 1. Graphs and Combinatorics. 16(4), 467–495 (2000)

    Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Xuechao Li.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Li, X. Sizes of Critical Graphs with Small Maximum Degrees. Graphs and Combinatorics 22, 503–513 (2006). https://doi.org/10.1007/s00373-006-0679-7

Download citation

  • Received:

  • Accepted:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00373-006-0679-7

Keywords

Navigation