Skip to main content

Volume 18 Supplement 2

GraphCliques

  • SI: GraphCliques
  • Open access
  • Published:

On the classification problem for split graphs

Abstract

The Classification Problem is the problem of deciding whether a simple graph has chromatic index equal to Δ or Δ+1. In the first case, the graphs are called Class 1, otherwise, they are Class 2. A split graph is a graph whose vertex set admits a partition into a stable set and a clique. Split graphs are a subclass of chordal graphs. Figueiredo at al. (J. Combin. Math. Combin. Comput. 32:79–91, 2000) state that a chordal graph is Class 2 if and only if it is neighborhood-overfull. In this paper, we give a characterization of neighborhood-overfull split graphs and we show that the above conjecture is true for some split graphs.

References

  1. Barbosa MM, de Mello CP, Meidanis J (1998) Local conditions for edge-colouring of cographs. Congr Numer 133:45–55

    MATH  MathSciNet  Google Scholar 

  2. Cai L, Ellis JL (1991) NP-completeness of edge-colouring some restricted graphs. Discrete Appl Math 30:15–27

    Article  MATH  MathSciNet  Google Scholar 

  3. Chen B-L, Fu H-L, Ko MT (1995) Total chromatic number and chromatic index of split graphs. J Comb Math Comb Comput 17:137–146

    MATH  MathSciNet  Google Scholar 

  4. Chetwynd AG, Hilton AJW (1986) Star multigraphs with three vertices of maximum degree. Math Proc Camb Philos Soc 100:300–317

    Article  MathSciNet  Google Scholar 

  5. Chetwynd AG, Hilton AJW (1989) The edge-chromatic class of graphs with even maximum degree at least |V|−3. Ann Discrete Math 41:91–110

    Article  MathSciNet  Google Scholar 

  6. Coneil DG, Perl Y (1984) Clustering and domination in perfect graphs. Discrete Appl Math 9(1):27–39

    Article  MathSciNet  Google Scholar 

  7. Cozzens MB, Halsey MD (1991) The relationship between the threshold dimension of split graphs and various dimensional parameters. Discrete Appl Math 30(2–3):125–135

    Article  MATH  MathSciNet  Google Scholar 

  8. Figueiredo CMH, Meidanis J, de Mello CP (2000) Local conditions for edge-coloring. J Comb Math Comb Comput 32:79–91

    MATH  Google Scholar 

  9. Figueiredo CMH, Meidanis J, de Mello CP (1999) Total-chromatic number and chromatic index of dually chordal graphs. Inf Process Lett 70:147–152

    Article  Google Scholar 

  10. Fournier JC (1973) Coloration des arêtes d’un graphe. Cah Cent étud Rech Opér 15:311–314

    MATH  MathSciNet  Google Scholar 

  11. Hilton AJW (1989) Two conjectures on edge-coloring. Discrete Math 74:61–64

    Article  MATH  MathSciNet  Google Scholar 

  12. Hilton AJW, Zhao C (1992) The chromatic index of a graph whose core has maximum degree two. Discrete Math 101:135–147

    Article  MATH  MathSciNet  Google Scholar 

  13. Hilton AJW, Zhao C (1996) On the edge-colouring of graphs whose core has maximum degree two. J Comb Math Comb Comput 21:97–108

    MATH  MathSciNet  Google Scholar 

  14. Hoffman DG, Rodger CA (1992) The chromatic index of complete multipartite graphs. J Graph Theory 16:159–163

    Article  MATH  MathSciNet  Google Scholar 

  15. Holyer I (1981) The NP-completeness of edge-coloring. SIAM J Comput 10:718–720

    Article  MATH  MathSciNet  Google Scholar 

  16. König D (1916) Über Graphen und ihre Anwendung auf Determinantentheorie und Mengenlehre. Math Ann 77:453–465

    Article  MATH  MathSciNet  Google Scholar 

  17. Machado RCS, Figueiredo CMH (2010) Decompositions for edge-coloring join graphs and cobipartite graphs. Discrete Appl Math 158(12):1336–1342

    Article  MATH  MathSciNet  Google Scholar 

  18. Makino K, Uno Y, Ibaraki T (2006) Minimum edge ranking spanning trees of split graphs. Discrete Appl Math 154(16):2373–2386

    Article  MATH  MathSciNet  Google Scholar 

  19. Ortiz C, Maculan N, Szwarcfiter JL (1998) Characterizing and edge-coloring split-indifference graphs. Discrete Appl Math 82:209–217

    Article  MATH  MathSciNet  Google Scholar 

  20. Plantholt MJ (1981) The chromatic index of graphs with a spanning star. J Graph Theory 5:45–53

    Article  MATH  MathSciNet  Google Scholar 

  21. Plantholt MJ (1983) The chromatic index of graphs with large maximum degree. Discrete Math 47:91–96

    Article  MATH  MathSciNet  Google Scholar 

  22. Royle GF (2000) Counting set covers and split graphs. J Integer Seq 3:1–5

    MathSciNet  Google Scholar 

  23. Sanders DP, Zhao Y (2001) Planar graphs of maximum degree seven are Class I. J Comb Theory, Ser B 83(2):201–212

    Article  MATH  MathSciNet  Google Scholar 

  24. De Simone C, de Mello CP (2006) Edge-colouring of join graphs. Theor Comput Sci 355(3):364–370

    Article  MATH  Google Scholar 

  25. De Simone C, Galluccio A (2007) Edge-colouring of regular graphs of large degree. Theor Comput Sci 389(1–2):91–99

    Article  MATH  Google Scholar 

  26. Tan ND, Hung LX (2006) On colorings of split graphs. Acta Math Vietnam 31(3):195–204

    MATH  MathSciNet  Google Scholar 

  27. Vizing VG (1964) On an estimate of the chromatic class of a p-graph. Diskretn Anal 3:25–30

    MathSciNet  Google Scholar 

  28. Vizing VG (1965) Critical graphs with a given chromatic class. Metody Diskretn Anal 5:9–17 (in Russian)

    MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Sheila Morais de Almeida.

Rights and permissions

Open Access This article is distributed under the terms of the Creative Commons Attribution 2.0 International License ( https://creativecommons.org/licenses/by/2.0 ), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Reprints and permissions

About this article

Cite this article

Morais de Almeida, S., Picinin de Mello, C. & Morgana, A. On the classification problem for split graphs. J Braz Comput Soc 18, 95–101 (2012). https://doi.org/10.1007/s13173-011-0046-2

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s13173-011-0046-2

Keywords