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Combinatorica

, Volume 38, Issue 5, pp 1021–1043 | Cite as

Chromatic Number of Ordered Graphs with Forbidden Ordered Subgraphs

  • Maria Axenovich
  • Jonathan Rollin
  • Torsten Ueckerdt
Original Paper
  • 26 Downloads

Abstract

It is well-known that the graphs not containing a given graph H as a subgraph have bounded chromatic number if and only if H is acyclic. Here we consider ordered graphs, i.e., graphs with a linear ordering ≺ on their vertex set, and the function
$${f_ \prec }\left( H \right) = \sup \left\{ {\chi \left( G \right)|G \in For{b_ \prec }\left( H \right)} \right\},$$
where Forb(H) denotes the set of all ordered graphs that do not contain a copy of H.

If H contains a cycle, then as in the case of unordered graphs, f(H)=∞. However, in contrast to the unordered graphs, we describe an infinite family of ordered forests H with f(H) =∞. An ordered graph is crossing if there are two edges uv and uv′ with uu′ ≺ vv′. For connected crossing ordered graphs H we reduce the problem of determining whether f(H) ≠∞ to a family of so-called monotonically alternating trees. For non-crossing H we prove that f(H) ≠∞ if and only if H is acyclic and does not contain a copy of any of the five special ordered forests on four or five vertices, which we call bonnets. For such forests H, we show that f(H)⩽2|V(H)| and that f(H)⩽2|V (H)|−3 if H is connected.

Mathematics Subject Classification (2000)

05C15 05C35 

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References

  1. [1]
    L. Addario-Berry, F. Havet, C. L. Sales, B. Reed and S. Thomassé: Oriented trees in digraphs, Discrete Math. 313 (2006), 967–974, 2013.MathSciNetCrossRefzbMATHGoogle Scholar
  2. [2]
    M. Balko, J. Cibulka, K. Král and J. Kynčl: Ramsey numbers of ordered graphs, Electronic Notes in Discrete Mathematics, 49:419–424, 2015. The Eighth European Conference on Combinatorics, Graph Theory and Applications, EuroComb 2015.Google Scholar
  3. [3]
    P. Brass, G. Károlyi and P. Valtr: A Turán-type extremal theory of convex geometric graphs, In Discrete and computational geometry, volume 25 of Algorithms Combin., pages 275–300. Springer, Berlin, 2003.CrossRefGoogle Scholar
  4. [4]
    S. A. Burr: Subtrees of directed graphs and hypergraphs, in: Proceedings of the Eleventh Southeastern Conference on Combinatorics, Graph Theory and Computing (Florida Atlantic Univ., Boca Raton, Fla., 1980), Vol. I, volume 28 of Congr. Numer., 227–239, 1980.Google Scholar
  5. [5]
    V. Capoyleas and J. Pach: A Turán-type theorem on chords of a convex polygon, J. Combin. Theory Ser. B 56 (2006), 9–15, 1992.MathSciNetCrossRefzbMATHGoogle Scholar
  6. [6]
    V. Chvátal: Perfectly ordered graphs, in: Topics on perfect graphs, volume 88 of North-Holland Math. Stud., pages 63–65. North-Holland, Amsterdam, 1984.CrossRefGoogle Scholar
  7. [7]
    D. Conlon, J. Fox, C. Lee and B. Sudakov: Ordered Ramsey numbers, J. Combin. Theory Ser. B. 122 (2017), 353–383.MathSciNetCrossRefzbMATHGoogle Scholar
  8. [8]
    P. Damaschke: Forbidden ordered subgraphs, in: Topics in combinatorics and graph theory (Oberwolfach, 1990), 219–229. Physica, Heidelberg, 1990.CrossRefGoogle Scholar
  9. [9]
    B. Descartes: A three colour problem, Eureka 21, 1947.Google Scholar
  10. [10]
    V. Dujmovic and D. R. Wood: On linear layouts of graphs, Discrete Math. Theor. Comput. Sci. 6 (2006), 339–358, 2004.MathSciNetzbMATHGoogle Scholar
  11. [11]
    P. Erdős: Graph theory and probability, Canad. J. Math. 11 (1959), 34–38.MathSciNetCrossRefzbMATHGoogle Scholar
  12. [12]
    Z. Füredi and P. Hajnal: Davenport-Schinzel theory of matrices. Discrete Math. 103 (2006), 233–251, 1992.MathSciNetCrossRefzbMATHGoogle Scholar
  13. [13]
    M. Ginn: Forbidden ordered subgraph vs. forbidden subgraph characterizations of graph classes, J. Graph Theory 30 (2006), 71–76, 1999.MathSciNetCrossRefzbMATHGoogle Scholar
  14. [14]
    M. Klazar: Extremal problems for ordered (hyper) graphs: applications of Davenport-Schinzel sequences, European J. Combin. 25 (2006), 125–140, 2004.MathSciNetCrossRefzbMATHGoogle Scholar
  15. [15]
    M. Klazar: Extremal problems for ordered hypergraphs: small patterns and some enumeration, Discrete Appl. Math. 143 (2006), 144–154, 2004.MathSciNetCrossRefzbMATHGoogle Scholar
  16. [16]
    A. Marcus and G. Tardos: Excluded permutation matrices and the Stanley-Wilf conjecture, J. Combin. Theory Ser. A 107 (2006), 153–160, 2004.MathSciNetCrossRefzbMATHGoogle Scholar
  17. [17]
    J. Pach and G. Tardos: Forbidden paths and cycles in ordered graphs and matrices, Israel J. Math. 155 (2006), 359–380.MathSciNetCrossRefzbMATHGoogle Scholar
  18. [18]
    G. Tardos: On 0–1 matrices and small excluded submatrices, J. Combin. Theory Ser. A 111 (2006), 266–288, 2005.MathSciNetCrossRefzbMATHGoogle Scholar
  19. [19]
    C. Weidert: Extremal problems in ordered graphs, Master’s thesis, Simon Fraser University, 2009. arXiv:0907.2479.Google Scholar
  20. [20]
    D. B. West: Introduction to graph theory, Prentice Hall, Inc., Upper Saddle River, NJ, 1996.zbMATHGoogle Scholar

Copyright information

© János Bolyai Mathematical Society and Springer-Verlag GmbH Germany, part of Springer Nature 2017

Authors and Affiliations

  • Maria Axenovich
    • 1
  • Jonathan Rollin
    • 1
  • Torsten Ueckerdt
    • 1
  1. 1.Department of MathematicsKarlsruhe Institute of TechnologyKarlsruheGermany

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