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Ramsey numbers for local colorings

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Abstract

The concept of a localk-coloring of a graphG is introduced and the corresponding localk-Ramsey numberr k loc (G) is considered. A localk-coloring ofG is a coloring of its edges in such a way that the edges incident to any vertex ofG are colored with at mostk colors. The numberr k loc (G) is the minimumm for whichK m contains a monochromatic copy ofG for every localk-coloring ofK m . The numberr k loc (G) is a natural generalization of the usual Ramsey numberr k (G) defined for usualk-colorings. The results reflect the relationship betweenr k (G) andr k loc (G) for certain classes of graphs.

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References

  1. Burr, S.A., Erdös, P.: Extremal Ramsey theory for graphs. Utilitas math.9, 247–258 (1976)

    Google Scholar 

  2. Burr, S.A., Erdös, P., Spencer, J.H.: Ramsey theorems for multiple copies of graphs. Trans. Amer. Math. Soc.209, 87–99 (1975)

    Google Scholar 

  3. Burr, S.A., Roberts, J.A.: On Ramsey numbers for stars. Utilitas Math.4, 217–220 (1973)

    Google Scholar 

  4. Chung, F.R.K.: On a Ramsey-type problem. J. Graph Theory.7, 79–83 (1983)

    Google Scholar 

  5. Cockayne, E.J., Lorimer, P.J.: The Ramsey graph numbers for stripes. J. Aust. Math. Soc.19, 252–256 (1975)

    Google Scholar 

  6. Erdös, P. Faudree, R.J., Rousseau, C.C., Schelp, R.H.: Ramsey numbers for brooms, Utilitas Math.35, 283–293 (1982)

    Google Scholar 

  7. Erdös, P., Lovász, L.: Problems and results on 3-chromatic hypergraphs and some related questions. Colloq. Math. Soc. Janos. Bolyai10, 609–627 (1974)

    Google Scholar 

  8. Faudree, R.J., Schelp, R.H.: All Ramsey numbers for cycles in graphs. Discrete Math.8, 313–329 (1974)

    Google Scholar 

  9. Gerencsér, L., Gyárfás, A.: On Ramsey type problems. Ann. Univ. Sci. Budap. Eötvös Eötvos Sect. Math.10, 167–170 (1967)

    Google Scholar 

  10. Greenwoood, R.E., Gleason, A.M.: Combinatorial relations and chromatic graphs. Can. J. Math.9, 1–7 (1955)

    Google Scholar 

  11. Irving, R.W.: Generalized Ramsey numbers for small graphs. Discrete Math.9, 251–264 (1974)

    Google Scholar 

  12. Ray-Chaudhuri, D.K. Wilson, R.M.: Solution of Kirkman's school girl problem. Proc. Symp. Pure Math.19, 187–204 (1972)

    Google Scholar 

  13. Rosta, V.: On a Ramsey type problem of J.A. Bondy and P. Erdös I, II. J. Comb. Theory (B)15, 94–120 (1973)

    Google Scholar 

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Authors and Affiliations

  1. Computer and Automation Institute, Hungarian Academy of Sciences, Kende u. 13-17, 1111, Budapest, Hungary

    A. Gyárfás, J. Lehel & ZS. Tuza

  2. Memphis State University, 38152, Memphis, TN, USA

    R. H. Schelp

Authors
  1. A. Gyárfás
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  2. J. Lehel
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  3. R. H. Schelp
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  4. ZS. Tuza
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This research was done while under an IREX grant.

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Gyárfás, A., Lehel, J., Schelp, R.H. et al. Ramsey numbers for local colorings. Graphs and Combinatorics 3, 267–277 (1987). https://doi.org/10.1007/BF01788549

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  • DOI: https://doi.org/10.1007/BF01788549

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