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## Chromatic number, clique subdivisions, and the conjectures of Hajos and Erdos-Fajtlowicz

##### Author(s)

Fox, Jacob; Lee, Choongbum; Sudakov, Benny
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Show full item record##### 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 .

##### Department

Massachusetts Institute of Technology. Department of Mathematics##### Journal

Combinatorica

##### Publisher

Springer-Verlag

##### Citation

Fox, Jacob, Choongbum Lee and Benny Sudakov. "Chromatic number, clique subdivisions, and the conjectures of Hajos and Erdos-Fajtlowicz." Combinatorica 32 (1) (January 2012) p. 111-123.

Version: Author's final manuscript

##### ISSN

0209-9683

1439-6912