This is not the latest version of this item. The latest version can be found here.
Chromatic number, clique subdivisions, and the conjectures of Hajos and Erdos-Fajtlowicz
Name
Fox_Chromatic number.pdf
Size
292.09 KB
Format
Adobe PDF
Checksum (MD5)
0300cfa4e169d8fafdbb2172f30d7d0c
Author(s)
Fox, Jacob
Lee, Choongbum
Sudakov, Benny
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
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
Terms of Use
Creative Commons Attribution-Noncommercial-Share Alike 3.0
Persistent DSpace Link
DOI of Published Version
http://dx.doi.org/10.1007/s00493-012-2709-9
