Tilings in Randomly Perturbed Dense Graphs

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 utilises 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.

Spanning trees in randomly perturbed graphs

“…Our approach is as follows. We first apply the regularity lemma to G to obtain a subgraph G ′ and a partition {V i } i∈[r]× [2] of V(G) such that G ′ [V (i,1) , V (i, 2) ] is ( , )-super-regular for all i ∈ [r]. We decompose T into subforests F 1 , … , F k+1 , F ′ 1 , … , F ′ k , L 1 and L last such that Δ(F) = O(np) for all F ∈ {F 1 , … , F k+1 , L 1 } and F ′ 1 , … , F ′ k , L 1 , L last are star-forests.…”

“…There are several further obstacles. For example, say, the centers x 1 , … , x s of L last are embedded into V i,1 for some i, the number ∑ i∈[s] L last (x i ) of leaves attached to the centers might not be equal to the number of vertices left in V i, 2 . In this case, it is impossible to find an exact embedding using (b1).…”

“…Furthermore, we will reserve a small subgraph F • of L last . Before we embed L last into G ′ , we embed exactly the right number of leaf-vertices of L 1 ∪ F • into each V i for each i ∈ [r] × [2] by using (a1). Hence this problem does not occur when we are about to embed L last ⧵ F • .…”

“…In Section 3, we introduce notation, state some probabilistic tools, and present some results involving graph regularity terminology and introduce a tightness example. In Section 4, we prove (a1)-(b1) and we prove Lemma 4.6 which we use in Section 6 to assign the vertices of T to the different sets in {V i } i∈[r]× [2] . In Section 5, we construct the (edge) decomposi-…”

“…We use this lemma later to assign subforests of T to different clusters of the regularity partition (see Section 6). Recall that for i = (i, h) ∈ N × [2], we write i for (i, 3 − h). To be a bit more precise, for a graph G, suppose we have a partition {V i } i∈[r]× [2] of V(G) as given by Lemma 3.10 and suppose we decide to embed a subtree…”