AbstractGiven 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 𝒢(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, χcr(H) denotes the critical chromatic number, a parameter introduced by Komlós, and m2(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
= K3, 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.
On Komlós’ tiling theorem in random graphs
