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A New Bound for the 2/3 Conjecture†

Published online by Cambridge University Press:  23 January 2013

DANIEL KRÁL' ,
CHUN-HUNG LIU ,
JEAN-SÉBASTIEN SERENI ,
PETER WHALEN  and
ZELEALEM B. YILMA
DANIEL KRÁL'
Affiliation:
Institute of Mathematics, DIMAP and Department of Computer Science, University of Warwick, Coventry CV4 7AL, UK (e-mail: D.Kral@warwick.ac.uk)
CHUN-HUNG LIU
Affiliation:
School of Mathematics, Georgia Institute of Technology, Atlanta, GA 30332, USA (e-mail: cliu87@math.gatech.edu, pwhalen3@math.gatech.edu)
JEAN-SÉBASTIEN SERENI
Affiliation:
CNRS (LORIA), Nancy, France (e-mail: sereni@kam.mff.cuni.cz)
PETER WHALEN
Affiliation:
School of Mathematics, Georgia Institute of Technology, Atlanta, GA 30332, USA (e-mail: cliu87@math.gatech.edu, pwhalen3@math.gatech.edu)
ZELEALEM B. YILMA
Affiliation:
LIAFA (Université Denis Diderot), Paris, France (e-mail: Zelealem.Yilma@liafa.jussieu.fr)
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Abstract

We show that any n-vertex complete graph with edges coloured with three colours contains a set of at most four vertices such that the number of the neighbours of these vertices in one of the colours is at least 2n/3. The previous best value, proved by Erdős, Faudree, Gould, Gyárfás, Rousseau and Schelp in 1989, is 22. It is conjectured that three vertices suffice.

Keywords

Primary 05C35Secondary 05C55
Type
Paper
Information
Combinatorics, Probability and Computing , Volume 22 , Issue 3 , May 2013 , pp. 384 - 393
Copyright
Copyright © Cambridge University Press 2013

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Footnotes

†

This work was done in the framework of LEA STRUCO.

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