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Non-Three-Colourable Common Graphs Exist

Published online by Cambridge University Press:  16 March 2012

HAMED HATAMI ,
JAN HLADKÝ ,
DANIEL KRÁL' ,
SERGUEI NORINE  and
ALEXANDER RAZBOROV
HAMED HATAMI
Affiliation:
School of Computer Science, McGill University, Montreal, Canada (e-mail: [email protected])
JAN HLADKÝ
Affiliation:
Department of Applied Mathematics, Faculty of Mathematics and Physics, Charles University, Malostranské náměstí 25, 118 00 Prague, Czech Republic and DIMAP, Department of Computer Science, University of Warwick, Coventry CV4 7AL, UK (e-mail: [email protected])
DANIEL KRÁL'
Affiliation:
Institute for Theoretical Computer Science, Faculty of Mathematics and Physics, Charles University, Malostranské náměstí 25, 118 00 Prague, Czech Republic (e-mail: [email protected])
SERGUEI NORINE
Affiliation:
Department of Mathematics, Princeton University, Princeton, NJ, USA (e-mail: [email protected])
ALEXANDER RAZBOROV
Affiliation:
Department of Computer Science, University of Chicago, IL, USA (e-mail: [email protected])
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Abstract

A graph H is called common if the sum of the number of copies of H in a graph G and the number in the complement of G is asymptotically minimized by taking G to be a random graph. Extending a conjecture of Erdős, Burr and Rosta conjectured that every graph is common. Thomason disproved both conjectures by showing that K4 is not common. It is now known that in fact the common graphs are very rare. Answering a question of Sidorenko and of Jagger, Št'ovíček and Thomason from 1996 we show that the 5-wheel is common. This provides the first example of a common graph that is not three-colourable.

Keywords

Primary 05C3505C55
Type
Paper
Information
Combinatorics, Probability and Computing , Volume 21 , Issue 5 , September 2012 , pp. 734 - 742
Copyright
Copyright © Cambridge University Press 2012

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