On Komlós’ tiling theorem in random graphs
Abstract Given graphs G and H , a family of vertex-disjoint copies of H in G is called an H-tiling . Conlon, Gowers, Samotij and Schacht showed that for a given graph H and a constant γ >0 , there exists C >0 such that if $p \ge C{n^{ - 1/{m_2}(H)}}$ , then asymptotically almost surely every s...
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Published in | Combinatorics, probability & computing Vol. 29; no. 1; pp. 113 - 127 |
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Main Authors | , |
Format | Journal Article |
Language | English |
Published |
Cambridge
Cambridge University Press
01.01.2020
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Subjects | |
Online Access | Get full text |
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Summary: | Abstract
Given graphs
G
and
H
, a family of vertex-disjoint copies of
H
in
G
is called an
H-tiling
. Conlon, Gowers, Samotij and Schacht showed that for a given graph
H
and a constant
γ
>0
, there exists
C
>0
such that if
$p \ge C{n^{ - 1/{m_2}(H)}}$
, then asymptotically almost surely every spanning subgraph
G
of the random graph
(
n, p
)
with minimum degree at least
$\delta (G) \ge (1 - \frac{1}{{{\chi _{{\rm{cr}}}(H)}} + \gamma )np$
contains an
H
-tiling that covers all but at most
γn
vertices. Here,
χ
cr(
H
)
denotes the
critical chromatic number
, a parameter introduced by Komlós, and
m
2
(
H
) is the 2
-density
of
H
. We show that this theorem can be bootstrapped to obtain an
H
-tiling covering all but at most
$\gamma {(C/p)^{{m_2}(H)}}$
vertices, which is strictly smaller when
$p \ge C{n^{ - 1/{m_2}(H)}}$
. In the case where
H
=
K
3
, this answers the question of Balogh, Lee and Samotij. Furthermore, for an arbitrary graph
H
we give an upper bound on
p
for which some leftover is unavoidable and a bound on the size of a largest
H
-tiling for
p
below this value. |
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ISSN: | 0963-5483 1469-2163 |
DOI: | 10.1017/S0963548319000129 |