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The Structure of Typical Eye-Free Graphs and a Turán-Type Result for Two Weighted Colours

Published online by Cambridge University Press:  31 July 2017

PETER KEEVASH  and
WILLIAM LOCHET
PETER KEEVASH
Affiliation:
Mathematical Institute, University of Oxford, Woodstock Road, Oxford OX2 6GG, UK (e-mail: keevash@maths.ox.ac.uk)
WILLIAM LOCHET
Affiliation:
Computer Science Department, Ecole Normale Supérieure de Lyon, 46 Allée d'Italie, 69364 Lyon, France (e-mail: william.lochet@ens-lyon.fr)
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Abstract

The (a,b)-eye is the graph Ia,b = Ka+b \ Kb obtained by deleting the edges of a clique of size b from a clique of size a+b. We show that for any a,b ≥ 2 and p ∈ (0,1), if we condition the random graph G ~ G(n,p) on having no induced copy of Ia,b, then with high probability G is close to an a-partite graph or the complement of a (b−1)-partite graph. Our proof uses the recently developed theory of hypergraph containers, and a stability result for an extremal problem with two weighted colours. We also apply the stability method to obtain an exact Turán-type result for this extremal problem.

Keywords

05C35
Type
Paper
Information
Combinatorics, Probability and Computing , Volume 26 , Issue 6 , November 2017 , pp. 886 - 910
Copyright
Copyright © Cambridge University Press 2017 

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