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.
Similar content being viewed by others
References
N. Alon: Explicit Ramsey graphs and orthonormal labelings, Electron. J. Combin. 1 (1994), R12.
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.
B. Bollobás: The chromatic number of random graphs, Combinatorica 8 (1988), 49–55.
B. Bollobás and P. A. Catlin: Topological cliques of random graphs, J. Combin. Theory Ser. B 30 (1981), 224–227.
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.
P. Catlin: Hajós’ graph-coloring conjecture: variations and counterexamples, J. Combin. Theory Ser. B 26 (1979), 268–274.
G. Dirac: A property of 4-chromatic graphs and some remarks on critical graphs, J. London Math. Soc. 27 (1952), 85–92.
P. Erdős and S. Fajtlowicz: On the conjecture of Hajós, Combinatorica 1 (1981), 141–143.
P. Erdős and E. Szemerédi: On a Ramsey type theorem, Period. Math. Hungar. 2 (1972), 295–299.
J. Fox and B. Sudakov: Density theorems for bipartite graphs and related Ramsey-type results, Combinatorica 29 (2009), 153–196.
J. Fox and B. Sudakov: Dependent random choice, Random Structures Algorithms 38 (2011), 1–32.
T. Gowers: A new proof of Szemerédi’s theorem for arithmetic progressions of length four, Geom. Funct. Anal. 8 (1998), 529–551.
J. Komlós and E. Szemerédi: Topological cliques in graphs. II, Combin. Probab. Comput. 5 (1996), 79–90.
A. V. Kostochka and V. Rödl: On graphs with small Ramsey numbers, J. Graph Theory 37 (2001), 198–204.
M. Krivelevich and B. Sudakov: Pseudo-random graphs, in: More Sets, Graphs and Numbers, Bolyai Society Mathematical Studies 15, Springer, 2006, 199–262.
D. Kühn and D. Osthus: Topological minors in graphs of large girth, J. Combin. Theory Ser. B 86 (2002), 364–380.
B. Sudakov: Few remarks on the Ramsey-Turán-type problems, J. Combin Theory Ser. B 88 (2003), 99–106.
B. Sudakov: A conjecture of Erdős on graph Ramsey numbers, Adv. Math. 227 (2011), 601–609.
C. Thomassen: Some remarks on Hajós’ conjecture, J. Combin. Theory Ser. B 93 (2005), 95–105.
Author information
Authors and Affiliations
Corresponding author
Additional information
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.
Rights and permissions
About this article
Cite this article
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
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00493-013-2853-x