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Graphs and Combinatorics

, Volume 13, Issue 2, pp 139–146 | Cite as

A Class of Edge Critical 4-Chromatic Graphs

  • Guantao Chen
  • Paul Erdős
  • András Gyárfás
  • R. H. Schelp
Article

Abstract

We consider several constructions of edge critical 4-chromatic graphs which can be written as the union of a bipartite graph and a matching. In particular we construct such a graph G with each of the following properties: G can be contracted to a given critical 4-chromatic graph; for each n ≥ 7, G has n vertices and three matching edges (it is also shown that such graphs must have at least \({{8n} \over 5}\) edges); G has arbitrary large girth.

Keywords

Bipartite Graph Chromatic Number Critical Graph Partite Class Matching Edge 
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.

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References

  1. 1.
    Abbott, H.L., Hare, D.R., Zhou, B.: Sparse color-critical graphs and hypergraphs with no short cycles. J. of Graph Theory 18, 373–388 (1994)Google Scholar
  2. 2.
    Descartes, B.: Solution to advanced problem No. 4526. American Mathematical Monthly 61, 352 (1954)CrossRefMathSciNetGoogle Scholar
  3. 3.
    Bondy, J.A., Murty, U.S.: Graph theory with applications. New York: American Elsevier Publishing 1976zbMATHGoogle Scholar
  4. 4.
    Erdős, P.: Problems and results in graph theory and combinatorial analysis. In Problémes Combinatoires et Théorie des Graphes, Colloques Internationaux du C.N.R.S. No. 260, 127–129 (Orsay, 1976)Google Scholar
  5. 5.
    Erdős, P., Faudree, R.J., Rousseau, C.C., Schelp, R.H.: On cycle-complete graph Ramsey numbers. J. Graph Theory 2, 53–64 (1978)Google Scholar
  6. 6.
    Erdős, P., Hajnal, A.: On chromatic number of graphs and set systems. Acta Math. Acad. Sci. Hung. 17, 61–99 (1966)Google Scholar
  7. 7.
    Gallai, T.: Kritische Graphen (2), Magyar Tudományos Akadémia Matematikai Kutató Intézetének Közleményei, 8, 165–192 (1963)zbMATHMathSciNetGoogle Scholar
  8. 8.
    Lovász, L.: On chromatic number of finite set systems. Acta Math. Acad. Sci. Hung. 19, 59–67 (1968)Google Scholar
  9. 9.
    Nielsen, F., Toft, B.: On a class of planar 4-chromatic graphs due to T. Gallai. In: Recent Advances in Graph Theory, pp. 425–430, Proceedings of the Symposium in Prague 1974, Academica Praha, Praha 1975Google Scholar
  10. 10.
    Nešetřil, J., Rödl, V.: A short proof for the existence of highly chromatic hypergraphs without short cycles. J. of Comb. Theory Ser. B 27, 225–227 (1979)Google Scholar
  11. 11.
    Toft, B.: On the maximal number of edges of k-chromatic graphs. Studia Sci. Math. Hung. 5, 461–470 (1970)MathSciNetGoogle Scholar
  12. 12.
    Rödl, V., Tuza, Zs.: On Color Critical Graphs. J. Comb. Theory Ser. B 38, 204–213 (1985)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag 1997

Authors and Affiliations

  • Guantao Chen
    • 1
  • Paul Erdős
    • 2
  • András Gyárfás
    • 3
  • R. H. Schelp
    • 4
  1. 1.Georgia State UniversityAtlantaUSA
  2. 2.Mathematical InstituteHungarian Academy of SciencesHungary
  3. 3.Computer and Automation Research Institute, Hungarian Academy of SciencesUniversity of MemphisUSA
  4. 4.Department of MathematicsUniversity of MemphisMemphisUSA

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