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Tilings in Randomly Perturbed Dense Graphs

Part of: Graph theory

Published online by Cambridge University Press:  16 July 2018

JÓZSEF BALOGH ,
ANDREW TREGLOWN  and
ADAM ZSOLT WAGNER
JÓZSEF BALOGH
Affiliation:
Department of Mathematical Sciences, University of Illinois at Urbana-Champaign, Urbana, Illinois 61801, USA (e-mail: jobal@math.uiuc.edu)
ANDREW TREGLOWN
Affiliation:
School of Mathematics, University of Birmingham, Edgbaston, Birmingham B15 2TT, UK (e-mail: a.c.treglown@bham.ac.uk)
ADAM ZSOLT WAGNER
Affiliation:
University of Illinois at Urbana-Champaign, Urbana, Illinois 61801, USA (e-mail: zawagne2@illinois.edu)
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Abstract

A perfect H-tiling in a graph G is a collection of vertex-disjoint copies of a graph H in G that together cover all the vertices in G. In this paper we investigate perfect H-tilings in a random graph model introduced by Bohman, Frieze and Martin [6] in which one starts with a dense graph and then adds m random edges to it. Specifically, for any fixed graph H, we determine the number of random edges required to add to an arbitrary graph of linear minimum degree in order to ensure the resulting graph contains a perfect H-tiling with high probability. Our proof utilizes Szemerédi's Regularity Lemma [29] as well as a special case of a result of Komlós [18] concerning almost perfect H-tilings in dense graphs.

MSC classification

Primary: 05C35: Extremal problems
Secondary: 05C70: Factorization, matching, partitioning, covering and packing 05C80: Random graphs
Type
Paper
Information
Combinatorics, Probability and Computing , Volume 28 , Issue 2 , March 2019 , pp. 159 - 176
Copyright
Copyright © Cambridge University Press 2018 

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Footnotes

†

Research partially supported by NSF grant DMS-1500121 and an Arnold O. Beckman Research Award (UIUC Campus Research Board 15006).

‡

Research supported by EPSRC grant EP/M016641/1.

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