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Graph Parameters and Ramsey Theory

  • Vadim Lozin
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10765)

Abstract

Ramsey’s Theorem tells us that there are exactly two minimal hereditary classes containing graphs with arbitrarily many vertices: the class of complete graphs and the class of edgeless graphs. In other words, Ramsey’s Theorem characterizes the graph vertex number in terms of minimal hereditary classes where this parameter is unbounded. In the present paper, we show that a similar Ramsey-type characterization is possible for a number of other graph parameters, including neighbourhood diversity and VC-dimension.

Notes

Acknowledgment

This work was supported by the Russian Science Foundation Grant No. 17-11-01336.

References

  1. 1.
    Alon, N., Brightwell, G., Kierstead, H., Kostochka, A., Winkler, P.: Dominating sets in \(k\)-majority tournaments. J. Combin. Theory Ser. B 96, 374–387 (2006)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Atminas, A., Lozin, V.V., Razgon, I.: Linear time algorithm for computing a small biclique in graphs without long induced paths. In: Fomin, F.V., Kaski, P. (eds.) SWAT 2012. LNCS, vol. 7357, pp. 142–152. Springer, Heidelberg (2012).  https://doi.org/10.1007/978-3-642-31155-0_13CrossRefzbMATHGoogle Scholar
  3. 3.
    Courcelle, B., Olariu, S.: Upper bounds to the clique width of graphs. Discrete Appl. Math. 101, 77–114 (2000)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Dabrowski, K.K., Demange, M., Lozin, V.V.: New results on maximum induced matchings in bipartite graphs and beyond. Theor. Comput. Sci. 478, 33–40 (2013)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Gargano, L., Rescigno, A.: Complexity of conflict-free colorings of graphs. Theor. Comput. Sci. 566, 39–49 (2015)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Hammer, P.L., Kelmans, A.K.: On universal threshold graphs. Comb. Probab. Comput. 3(3), 327–344 (1994)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Lampis, M.: Algorithmic meta-theorems for restrictions of treewidth. Algorithmica 64(1), 19–37 (2012)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Lin, B.: The parameterized complexity of k-Biclique. In: Proceedings of the Twenty-Sixth Annual ACM-SIAM Symposium on Discrete Algorithms, pp. 605–615 (2015)Google Scholar
  9. 9.
    Lozin, V., Rudolf, G.: Minimal universal bipartite graphs. Ars Comb. 84, 345–356 (2007)MathSciNetzbMATHGoogle Scholar
  10. 10.
    Lozin, V.: Boundary classes of planar graphs. Comb. Probab. Comput. 17(2), 287–295 (2008)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Lozin, V.: Minimal classes of graphs of unbounded clique-width. Ann. Comb. 15(4), 707–722 (2011)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Ramsey, F.P.: On a problem of formal logic. Proc. Lond. Math. Soc. 30, 264–286 (1930)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Robertson, N., Seymour, P.D.: Graph minors. V. Excluding a planar graph. J. Comb. Theory Ser. B 41, 92–114 (1986)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Mathematics InstituteUniversity of WarwickCoventryUK
  2. 2.Lobachevsky State University of Nizhni NovgorodNizhny NovgorodRussia

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