Chromatic number, clique subdivisions, and the conjectures of Hajós and Erdős-Fajtlowicz

Abstract

For a graph G, let χ(G) denote its chromatic number and σ(G) denote the order of the largest clique subdivision in G. Let H(n) be the maximum of χ(G)=σ(G) over all n-vertex graphs G. A famous conjecture of Hajós from 1961 states that σ(G) ≥ χ(G) for every graph G. That is, H(n)≤1 for all positive integers n. This conjecture was disproved by Catlin in 1979. Erdős and Fajtlowicz further showed by considering a random graph that H(n)≥cn 1/2/logn for some absolute constant c>0. In 1981 they conjectured that this bound is tight up to a constant factor in that there is some absolute constant C such that χ(G)=σ(G) ≤ Cn 1/2/logn for all n-vertex graphs G. In this paper we prove the Erdős-Fajtlowicz conjecture. The main ingredient in our proof, which might be of independent interest, is an estimate on the order of the largest clique subdivision which one can find in every graph on n vertices with independence number α.

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References

  1. [1]

    N. Alon: Explicit Ramsey graphs and orthonormal labelings, Electron. J. Combin. 1 (1994), R12.

    Google Scholar 

  2. [2]

    N. Alon, M. Krivelevich and B. Sudakov: Turan numbers of bipartite graphs and related Ramsey-type questions, Combinatorics, Probability and Computing 12 (2003), 477–494.

    MathSciNet  MATH  Article  Google Scholar 

  3. [3]

    B. Bollobás: The chromatic number of random graphs, Combinatorica 8 (1988), 49–55.

    MathSciNet  MATH  Article  Google Scholar 

  4. [4]

    B. Bollobás and P. A. Catlin: Topological cliques of random graphs, J. Combin. Theory Ser. B 30 (1981), 224–227.

    MathSciNet  MATH  Article  Google Scholar 

  5. [5]

    B. Bollobás and A. Thomason: Proof of a conjecture of Mader, Erdős and Hajnal on topological complete subgraphs, European J. Combin. 19 (1998), 883–887.

    MathSciNet  MATH  Article  Google Scholar 

  6. [6]

    P. Catlin: Hajós’ graph-coloring conjecture: variations and counterexamples, J. Combin. Theory Ser. B 26 (1979), 268–274.

    MathSciNet  MATH  Article  Google Scholar 

  7. [7]

    G. Dirac: A property of 4-chromatic graphs and some remarks on critical graphs, J. London Math. Soc. 27 (1952), 85–92.

    MathSciNet  MATH  Article  Google Scholar 

  8. [8]

    P. Erdős and S. Fajtlowicz: On the conjecture of Hajós, Combinatorica 1 (1981), 141–143.

    MathSciNet  Article  Google Scholar 

  9. [9]

    P. Erdős and E. Szemerédi: On a Ramsey type theorem, Period. Math. Hungar. 2 (1972), 295–299.

    MathSciNet  Article  Google Scholar 

  10. [10]

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

    MathSciNet  MATH  Google Scholar 

  11. [11]

    J. Fox and B. Sudakov: Dependent random choice, Random Structures Algorithms 38 (2011), 1–32.

    MathSciNet  Article  Google Scholar 

  12. [12]

    T. Gowers: A new proof of Szemerédi’s theorem for arithmetic progressions of length four, Geom. Funct. Anal. 8 (1998), 529–551.

    MathSciNet  MATH  Article  Google Scholar 

  13. [13]

    J. Komlós and E. Szemerédi: Topological cliques in graphs. II, Combin. Probab. Comput. 5 (1996), 79–90.

    MathSciNet  MATH  Article  Google Scholar 

  14. [14]

    A. V. Kostochka and V. Rödl: On graphs with small Ramsey numbers, J. Graph Theory 37 (2001), 198–204.

    MathSciNet  MATH  Article  Google Scholar 

  15. [15]

    M. Krivelevich and B. Sudakov: Pseudo-random graphs, in: More Sets, Graphs and Numbers, Bolyai Society Mathematical Studies 15, Springer, 2006, 199–262.

    Google Scholar 

  16. [16]

    D. Kühn and D. Osthus: Topological minors in graphs of large girth, J. Combin. Theory Ser. B 86 (2002), 364–380.

    MathSciNet  MATH  Article  Google Scholar 

  17. [17]

    B. Sudakov: Few remarks on the Ramsey-Turán-type problems, J. Combin Theory Ser. B 88 (2003), 99–106.

    MathSciNet  MATH  Article  Google Scholar 

  18. [18]

    B. Sudakov: A conjecture of Erdős on graph Ramsey numbers, Adv. Math. 227 (2011), 601–609.

    MathSciNet  MATH  Article  Google Scholar 

  19. [19]

    C. Thomassen: Some remarks on Hajós’ conjecture, J. Combin. Theory Ser. B 93 (2005), 95–105.

    MathSciNet  MATH  Article  Google Scholar 

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Correspondence to Jacob Fox.

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Research supported by a Simons Fellowship.

Research supported in part by a Samsung Scholarship.

Research supported in part by NSF grant DMS-1101185, by AFOSR MURI grant FA9550-10-1-0569 and by a USA-Israel BSF grant.

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Fox, J., Lee, C. & Sudakov, B. Chromatic number, clique subdivisions, and the conjectures of Hajós and Erdős-Fajtlowicz. Combinatorica 33, 181–197 (2013). https://doi.org/10.1007/s00493-013-2853-x

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Mathematics Subject Classi cation (2000)

  • 05D10
  • 05C55