Chromatic number, clique subdivisions, and the conjectures of Hajos and Erdos-Fajtlowicz
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Fox_Chromatic number.pdf
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Author(s)
Fox, Jacob
Lee, Choongbum
Sudakov, Benny
Date Issued
February 2012
Journal
Combinatorica
Publisher
Springer-Verlag
Citation
Fox, Jacob et al. "Chromatic number, clique subdivisions, and the conjectures of Hajos and Erdos-Fajtlowicz." Combinatorica 32, 1 (January 2012): 111-123.
Version
Author's final manuscript
Abstract
For a graph G, let (G) denote its chromatic number and (G) denote the order of the largest
clique subdivision in G. Let H(n) be the maximum of (G)= (G) over all n-vertex graphs G.
A famous conjecture of Haj os from 1961 states that (G) (G) for every graph G. That is,
H(n) 1 for all positive integers n. This conjecture was disproved by Catlin in 1979. Erd}os
and Fajtlowicz further showed by considering a random graph that H(n) cn1=2= log n 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 (G)= (G) Cn1=2= log n for all n-vertex
graphs G. In this paper we prove the Erd}os-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 nd in every graph on n vertices with independence number .
MIT Department
Massachusetts Institute of Technology. Department of Mathematics
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DOI of Published Version
https://doi.org/10.1007/s00493-012-2709-9
