Monotone paths in ordered graphs

Abstract

LetV fin andE fin resp. denote the classes of graphsG with the property that no matter how we label the vertices (edges, resp.) ofG by members of a linearly ordered set, there will exist paths of arbitrary finite lengths with monotonically increasing labels. The classesV inf andE inf are defined similarly by requiring the existence of an infinite path with increasing labels. We proveE infV infV finE fin. Finally we consider labellings by positive integers and characterize the class corresponding toV inf.

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References

  1. [1]

    N. G. de Bruijn andP. Erdős, A color problem for infinite graphs and a problem in the theory of relations,Indag. Math. 13 (1951), 369–373.

    Google Scholar 

  2. [2]

    T. Gallai, On directed paths and circuits, in:Theory of Graphs (P. Erdös and G. Katona, eds.), Academic Press 1968.

  3. [3]

    R. L. Graham andD. J. Kleitman, Increasing paths in edge ordered graphs,Periodica Mathematica Hungarica 3 (1973), 141–148.

    MATH  Article  MathSciNet  Google Scholar 

  4. [4]

    F. P. Ramsey, On a problem of formal logic,Proc. Lond. Math. Soc. (2)30 (1930), 264–286.

    Article  Google Scholar 

  5. [5]

    V. Rödl, Generalization of the Ramsey Theorem and Dimension of Graphs, (in Czech),Thesis, Charles University, Prague (1973).

    Google Scholar 

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Müller, V., Rödl, V. Monotone paths in ordered graphs. Combinatorica 2, 193–201 (1982). https://doi.org/10.1007/BF02579318

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AMS subject classification (1980)

  • 05 C 55
  • 05 C 38