Classes of graphs that exclude a tree and a clique and are not vertex ramsey

Abstract

A class Γ of graphs is vertex Ramsey if for allH∈Γ there existsG∈Γ such that for all partitions of the vertices ofG into two parts, one of the parts contains an induced copy ofH. Forb (T,K) is the class of graphs that induce neitherT norK. LetT(k, r) be the tree with radiusr such that each nonleaf is adjacent tok vertices farther from the root than itself. Gyárfás conjectured that for all treesT and cliquesK, there exists an integerb such that for allG in Forb(T,K), the chromatic number ofG is at mostb. Gyárfás' conjecture implies a weaker conjecture of Sauer that for all treesT and cliquesK, Forb(T,K) is not vertex Ramsey. We use techniques developed for attacking Gyárfás' conjecture to prove that for allq, r and sufficiently largek, Forb(T(k,r),K q ) is not vertex Ramsey.

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References

  1. [1]

    A. Gyárfás:On Ramsey Covering-Numbers, Coll. Math. Soc. János Bolyai 10, Infinite and Finite Sets North-Holland/American Elsevier, New York (1975), 801–816.

    Google Scholar 

  2. [2]

    A. Gyárfás: Problems from the world surrounding perfect graphs,Zastowania Matematyki Applicaaones Mathemacticae XIX, (1985), 413–441.

    Google Scholar 

  3. [3]

    A. Gyárfás, E. Szemerédi, andZs. Tuza: Induced subtrees in graphs of large chromatic number,Discrete Math.,30, (1980), 235–244.

    Google Scholar 

  4. [4]

    H. Kierstead: Long stars specify weaklyX-bounded classes,Colloquia Mathematica Societatis János Bolyai 60 Sets Graphs and Numbers, (1991), 421–428.

  5. [5]

    H. Kierstead: A class of graphs which is not vertex Ramsey,SIAM J. Discrete Math., to appear.

  6. [6]

    H. Kierstead, andS. Penrice: Radius two trees specifyX-bounded classes,J. of Graph Theory, (1994), 119–129.

  7. [7]

    H. Kierstead, andS. Penrice: Recent results on a conjecture of Gyárfás,Congressus Num.,79 (1990), 182–186.

    Google Scholar 

  8. [8]

    H. Kierstead, S. Penrice, andW. T. Trotter: On-line graph coloring and recursive graph theory,SIAM J. on Discrete Math,7 (1994), 72–89.

    Google Scholar 

  9. [9]

    H. Kierstead, andV. Rödl: Applications of hypergraph coloring to coloring graphs which do not induce certain trees,Discrete Mathematics,150 (1996), 187–193.

    Google Scholar 

  10. [10]

    V. Rödl, andN. Sauer: The Ramsey property of graphs which exclude a given graph,Can. J. Math.,44 (1992), 1050–1060.

    Google Scholar 

  11. [11]

    V. Rödl, N. Sauer, andX. Zhu: Ramsey families which exclude a graph,Combinatorica,15 (1995), 589–596.

    Google Scholar 

  12. [12]

    N. Sauer: Vertex partition problems, Combinatorics, Paul Erdős Is Eighty (Volume I), (1993), 361–377.

    Google Scholar 

  13. [13]

    N. Sauer: On the Ramsey property of families of graphs,Transactions of A. M. S., to appear.

  14. [14]

    N. Sauer, andX. Zhu: Graphs that do not embed a given graph and the Ramsey Property,Colloquia Mathematica Societatis János Bolyai 60, Sets Graphs and Numbers, (1991) 631–636.

  15. [15]

    A. D. Scott: Induced trees in graphs of large chromatic number, (1993), manuscript.

  16. [16]

    D. P. Sumner: Subtrees of a graph and chromatic number, in:The Theory and Applications of Graphs, (ed. Gary Chartrand) John Wiley & Sons, New York (1981), 557–576.

    Google Scholar 

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Research partially supported by Office of Naval Research grant N00014-90-J-1206.

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Kierstead, H.A., Zhu, Y. Classes of graphs that exclude a tree and a clique and are not vertex ramsey. Combinatorica 16, 493–504 (1996). https://doi.org/10.1007/BF01271268

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Mathematics Subject Classification (1991)

  • 05 C