Ramsey-goodness—and otherwise

Abstract

A celebrated result of Chvátal, Rödl, Szemerédi and Trotter states (in slightly weakened form) that, for every natural number Δ, there is a constant r Δ such that, for any connected n-vertex graph G with maximum degree Δ, the Ramsey number R(G,G) is at most r Δ n, provided n is sufficiently large.

In 1987, Burr made a strong conjecture implying that one may take r Δ = Δ. However, Graham, Rödl and Ruciński showed, by taking G to be a suitable expander graph, that necessarily r Δ > 2 for some constant c>0. We show that the use of expanders is essential: if we impose the additional restriction that the bandwidth of G be at most some function β(n)=o(n), then R(G,G)≤(2χ(G)+4)n≤(2Δ+6)n, i.e., r Δ =2Δ+6 suffices. On the other hand, we show that Burr’s conjecture itself fails even for P k n , the kth power of a path P n .

Brandt showed that for any c, if Δ is sufficiently large, there are connected n-vertex graphs G with Δ(G)≤Δ but R(G,K 3) > cn. We show that, given Δ and H, there are β>0 and n 0 such that, if G is a connected graph on nn 0 vertices with maximum degree at most Δ and bandwidth at most β n , then we have R(G,H)=(χ(H)−1)(n−1)+σ(H), where σ(H) is the smallest size of any part in any χ(H)-partition of H. We also show that the same conclusion holds without any restriction on the maximum degree of G if the bandwidth of G is at most ɛ(H) log n=log logn.

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References

  1. [1]

    P. Allen: Covering two-edge-coloured complete graphs with two disjoint monochromatic cycles, Combin. Probab. Comput. 17 (2008), 471–486.

    MathSciNet  MATH  Google Scholar 

  2. [2]

    K. Appel and W. Haken: Every planar map is four colorable. Part I. Discharging, Illinois J. Math. 21 (1977), 429–490.

    MathSciNet  MATH  Google Scholar 

  3. [3]

    K. Appel, W. Haken and J. Koch: Every planar map is four colorable. Part II. Reducibility, Illinois J. Math. 21 (1977), 491–567.

    MathSciNet  MATH  Google Scholar 

  4. [4]

    J. A. Bondy, P. Erdős: Ramsey numbers for cycles in graphs, J. Combinatorial Theory Ser. B 14 (1973), 46–54.

    MathSciNet  MATH  Article  Google Scholar 

  5. [5]

    J. Böttcher, K. Pruessman, A. Taraz and A. Würfl: Bandwidth, expansion, treewidth, separators, and universality for bounded degree graphs, Eur. J. Comb. 31 (2010), 1217–1227.

    MATH  Article  Google Scholar 

  6. [6]

    J. Böttcher, M. Schacht and A. Taraz: Proof of the bandwidth conjecture of Bollobás and Komlós, Mathematische Annalen 343(1) (2009), 175–205.

    Article  Google Scholar 

  7. [7]

    S. Brandt: Expanding graphs and Ramsey numbers, available at Freie Universitäat, Berlin preprint server, ftp://ftp.math.fu-berlin.de/pub/math/publ/pre/1996/pr-a-96-24.ps (1996).

    Google Scholar 

  8. [8]

    W. G. Brown: On graphs that do not contain a Thomsen graph, Canad. Math. Bull. 9 (1966), 281–285.

    MathSciNet  MATH  Article  Google Scholar 

  9. [9]

    S. A. Burr: Ramsey numbers involving graphs with long suspended paths, J. London Math. Soc. 24 (1981), 405–413.

    MathSciNet  MATH  Article  Google Scholar 

  10. [10]

    S. A. Burr: What can we hope to accomplish in generalized Ramsey theory?, Discrete Math. 67 (1987), 215–225.

    MathSciNet  MATH  Article  Google Scholar 

  11. [11]

    S. A. Burr and P. Erdős: Generalizations of a Ramsey-theoretic result of Chvátal, J. Graph Theory 7 (1983), 39–51.

    MathSciNet  MATH  Article  Google Scholar 

  12. [12]

    S. A. Burr, P. Erdős, R. J. Faudree, C. C. Rousseau, R. H. Schelp: The Ramsey number for the pair complete bipartite graph-graph of limited degree, Graph theory with applications to algorithms and computer science (Kalamazoo, Mich., 1984), 163–174, Wiley, New York, 1985.

    Google Scholar 

  13. [13]

    G. Chen and R. H. Schelp: Graphs with linearly bounded Ramsey numbers, J. Combin. Theory Ser. B 57 (1993), 138–149.

    MathSciNet  MATH  Article  Google Scholar 

  14. [14]

    V. Chvátal: Tree-complete graph Ramsey number, J. Graph Theory 1 (1977), 93.

    MathSciNet  Article  Google Scholar 

  15. [15]

    V. Chvátal and F. Harary: Generalized Ramsey theory for graphs, III. Small off-diagonal numbers, Pacific J. Math 41 (1972), 335–345.

    MathSciNet  MATH  Article  Google Scholar 

  16. [16]

    V. Chvátal, V. Rödl, E. Szemerédi and W. T. Trotter: The Ramsey number of a graph with a bounded maximum degree, J. Combin. Theory Ser. B 34 (1983) 239–243.

    MathSciNet  MATH  Article  Google Scholar 

  17. [17]

    D. Conlon: Hypergraph packing and sparse bipartite Ramsey numbers, Combin. Probab. Comput. 18 (2009), 913–923.

    MathSciNet  MATH  Article  Google Scholar 

  18. [18]

    D. Conlon, J. Fox and B. Sudakov: On two problems in graph Ramsey theory, Combinatorica 32 (2012), 513–535.

    MathSciNet  Article  Google Scholar 

  19. [19]

    R. Diestel: Graph theory, third ed., Graduate Texts in Mathematics, vol. 173, Springer-Verlag, Berlin, 2005.

    Google Scholar 

  20. [20]

    P. Erdős: Some remarks on the theory of graphs, Bull. Amer. Math. Soc 53 (1947), 292–294.

    MathSciNet  Article  Google Scholar 

  21. [21]

    P. Erdős, R. J. Faudree, C. C. Rousseau and R. H. Schelp: On cycle-complete graph Ramsey numbers, J. Graph Theory 2 (1978), 53–64.

    MathSciNet  Article  Google Scholar 

  22. [22]

    P. Erdős and T. Gallai: On maximal paths and circuits of graphs, Acta Math. Acad. Sci. Hungar. 10 (1959), 337–356.

    MathSciNet  Article  Google Scholar 

  23. [23]

    P. Erdős and G. Szekeres: A combinatorial problem in geometry, Composito Math. 2 (1935), 464–470.

    Google Scholar 

  24. [24]

    R. J. Faudree and R. H. Schelp: All Ramsey numbers for cycles in graphs, Discrete Math. 8 (1974), 313–329.

    MathSciNet  MATH  Article  Google Scholar 

  25. [25]

    J. Fox and B. Sudakov: Density theorems for bipartite graphs and related Ramseytype results, Combinatorica 29 (2009), 153–196.

    MathSciNet  MATH  Google Scholar 

  26. [26]

    L. Gerencsér and A. Gyárfás: On Ramsey-type problems, Annales Universitatis Scientiarum Budapestinensis, Eötvös Sect. Math. 10 (1967), 167–170.

    MATH  Google Scholar 

  27. [27]

    R. L. Graham, V. Rödl and A. Ruciński: On graphs with linear Ramsey numbers, J. Graph Theory 35 (2000), 176–192.

    MathSciNet  MATH  Article  Google Scholar 

  28. [28]

    H. A. Kierstead and W. T. Trotter: Planar graph coloring with an uncooperative partner, J. Graph Theory 18 (1994), 569–584.

    MathSciNet  MATH  Article  Google Scholar 

  29. [29]

    Y. Kohayakawa, M. Simonovits and J. Skokan: The 3-coloured Ramsey number of odd cycles, J. Combin. Theory Ser. B, to appear.

  30. [30]

    Y. Kohayakawa, M. Simonovits and J. Skokan: Stability of Ramsey numbers for cycles, manuscript, 2008.

    Google Scholar 

  31. [31]

    J. Komlós, G. N. Sárközy and E. Szemerédi: Blow-up lemma, Combinatorica 17 (1997), 109–123.

    MathSciNet  MATH  Article  Google Scholar 

  32. [32]

    T. Kővári, V. Sós and P. Turán: On a problem of K. Zarankiewicz, Colloquium Math. 3 (1954), 50–57.

    MathSciNet  Google Scholar 

  33. [33]

    V. Nikiforov: The cycle-complete graph Ramsey numbers, Combin. Probab. Comput. 14 (2005), 349–370.

    MathSciNet  MATH  Article  Google Scholar 

  34. [34]

    V. Nikiforov and C. C. Rousseau: Ramsey goodness and beyond, Combinatorica 29 (2009), 227–262.

    MathSciNet  MATH  Google Scholar 

  35. [35]

    S. P. Radziszowski: Small Ramsey numbers, Electronic J. Combin DS1 (2006), 60pp.

  36. [36]

    V. Rosta: On a Ramsey-type problem of J. A. Bondy and P. Erdős. I, II, J. Combin. Theory Ser. B 15 (1973), 94–105 and 105–120.

    MathSciNet  MATH  Article  Google Scholar 

  37. [37]

    G. Sárközy, M. Schacht and A. Taraz: Two and three colour Ramsey numbers for bipartite graphs with small bandwidth, in preparation.

  38. [38]

    N. Sauer and J. Spencer: Edge disjoint placement of graphs, J. Combin. Theory Ser. B 25 (1978), 295–302.

    MathSciNet  MATH  Article  Google Scholar 

  39. [39]

    J. Spencer: Asymptotic lower bounds for Ramsey functions, Discrete Math. 20 (1977), 69–76.

    MathSciNet  Article  Google Scholar 

  40. [40]

    E. Szemerédi: Regular partitions of graphs, Colloques Internationaux C.N.R.S. Vol. 260, in Problémes Combinatoires et Théorie des Graphes, Orsay (1976), 399–401.

    Google Scholar 

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Correspondence to Peter Allen.

Additional information

During this work PA was supported by DIMAP and Mathematics Institute, University of Warwick, U.K., EPSRC award EP/D063191/1.

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Allen, P., Brightwell, G. & Skokan, J. Ramsey-goodness—and otherwise. Combinatorica 33, 125–160 (2013). https://doi.org/10.1007/s00493-013-2778-4

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Mathematics Subject Classification (2000)

  • 05C55
  • 05C35