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Graphs and Combinatorics

, Volume 3, Issue 1, pp 67–73 | Cite as

Linear upper bounds for local Ramsey numbers

  • Miroslaw Truszczynski
  • Zsolt Tuza
Article

Abstract

A coloring of the edges of a graph is called alocal k-coloring if every vertex is incident to edges of at mostk distinct colors. For a given graphG, thelocal Ramsey number, r loc k (G), is the smallest integern such that any localk-coloring ofK n , (the complete graph onn vertices), contains a monochromatic copy ofG. The following conjecture of Gyárfás et al. is proved here: for each positive integerk there exists a constantc = c(k) such thatr loc k (G) ≤ cr k (G), for every connected grraphG (wherer k (G) is theusual Ramsey number fork colors). Possible generalizations for hypergraphs are considered.

Keywords

Complete Graph Distinct Color Ramsey Number Monochromatic Copy Number Fork 
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

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    Chvátal, V., Rödl, V., Szemerédi, E., Trotter, W.T.: The Ramsey number of a graph with bounded maximum degree. J. Comb. Theory (B)34, 239–243 (1983)Google Scholar
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    Erdös, P., Kleitman, D.J.: On coloring graphs to maximize the proportion of multicoloredk-edges. J. Comb. Theory5, 164–169 (1968)Google Scholar
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    Gyárfás, A., Lehel, J., Nešetřil, J., Rödl, V., Schelp, R.H., Tuza, Zs.: Localk-colorings of graphs and hypergraphs. J. Comb. Theory (B) (to appear).Google Scholar
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    Gyárfás, A., Lehel, J., Schelp, R.H., Tuza, Zs.: Ramsey numbers for local colorings. Graphs and Combinatorics (to appear)Google Scholar

Copyright information

© Springer-Verlag 1987

Authors and Affiliations

  • Miroslaw Truszczynski
    • 1
  • Zsolt Tuza
    • 2
  1. 1.Department of Computer ScienceUniversity of KentuckyLexingtonUSA
  2. 2.Computer and Automation InstituteHungarian Academy of SciencesBudapestHungary

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