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Corrádi and Hajnal's Theorem for Sparse Random Graphs

Published online by Cambridge University Press:  02 February 2012

JÓZSEF BALOGH ,
CHOONGBUM LEE  and
WOJCIECH SAMOTIJ
JÓZSEF BALOGH
Affiliation:
Department of Mathematics, University of Illinois, 1409 W Green Street, Urbana, IL 61801, USA and Department of Mathematics, University of California, San Diego, CA 92093, USA (e-mail: jobal@math.uiuc.edu)
CHOONGBUM LEE
Affiliation:
Department of Mathematics, UCLA, Los Angeles, CA 90095, USA (e-mail: choongbum.lee@gmail.com)
WOJCIECH SAMOTIJ
Affiliation:
School of Mathematical Sciences, Tel Aviv University, Tel Aviv 69978, Israel and Trinity College, Cambridge CB2 1TQ, UK (e-mail: samotij@post.tau.ac.il)
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Abstract

In this paper we extend a classical theorem of Corrádi and Hajnal into the setting of sparse random graphs. We show that if p(n) ≫ (log n/n)1/2, then asymptotically almost surely every subgraph of G(n, p) with minimum degree at least (2/3 + o(1))np contains a triangle packing that covers all but at most O(p−2) vertices. Moreover, the assumption on p is optimal up to the (log n)1/2 factor and the presence of the set of O(p−2) uncovered vertices is indispensable. The main ingredient in the proof, which might be of independent interest, is an embedding theorem which says that if one imposes certain natural regularity conditions on all three pairs in a balanced 3-partite graph, then this graph contains a perfect triangle packing.

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Type
Paper
Information
Combinatorics, Probability and Computing , Volume 21 , Issue 1-2: Honouring the Memory of Richard H. Schelp , March 2012 , pp. 23 - 55
Copyright
Copyright © Cambridge University Press 2012

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