Tilings in randomly perturbed dense graphs

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 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 Szemer\'edi's Regularity lemma as well as a special case of a result of Koml\'os concerning almost perfect $H$-tilings in dense graphs.Comment: 19 pages, to appear in CP

Two conjectures in Ramsey-Tur\'an theory

Given graphs $H_1,\ldots, H_k$, a graph $G$ is $(H_1,\ldots, H_k)$-free if there is a $k$-edge-colouring $\phi:E(G)\rightarrow [k]$ with no monochromatic copy of $H_i$ with edges of colour $i$ for each $i\in[k]$. Fix a function $f(n)$, the Ramsey-Tur\'an function $\textrm{RT}(n,H_1,\ldots,H_k,f(n))$ is the maximum number of edges in an $n$-vertex $(H_1,\ldots,H_k)$-free graph with independence number at most $f(n)$. We determine $\textrm{RT}(n,K_3,K_s,\delta n)$ for $s\in\{3,4,5\}$ and sufficiently small $\delta$, confirming a conjecture of Erd\H{o}s and S\'os from 1979. It is known that $\textrm{RT}(n,K_8,f(n))$ has a phase transition at $f(n)=\Theta(\sqrt{n\log n})$. However, the values of $\textrm{RT}(n,K_8, o(\sqrt{n\log n}))$ was not known. We determined this value by proving $\textrm{RT}(n,K_8,o(\sqrt{n\log n}))=\frac{n^2}{4}+o(n^2)$, answering a question of Balogh, Hu and Simonovits. The proofs utilise, among others, dependent random choice and results from graph packings.Comment: 20 pages, 2 figures, 2 pages appendi

Two conjectures in Ramsey-Turan theory

Given graphs H 1 ,...,H k , a graph G is ( H 1 ,...,H k )-free if there is a k -edge-colouring φ : E ( G ) → [ k ] with no monochromatic copy of H i with edges of colour i for each i ∈ [ k ]. Fix a function f ( n ), the Ramsey-Tur ́an function RT( n,H 1 ,...,H k ,f ( n )) is the maximum number of edges in an n -vertex ( H 1 ,...,H k )-free graph with independence number at most f ( n ). We determine RT( n,K 3 ,K s ,δn ) for s ∈ { 3 , 4 , 5 } and sufficiently small δ , confirming a conjecture of Erd ̋os and S ́os from 1979. It is known that RT( n,K 8 ,f ( n )) has a phase transition at f ( n ) = Θ( √ n log n ). However, the value of RT( n,K 8 ,o ( √ n log n )) was not known. We determined this value by proving RT( n,K 8 ,o ( √ n log n )) = n 2 4 + o ( n 2 ), answering a question of Balogh, Hu and Simonovits. The proofs utilise, among others, dependent random choice and results from graph packings

The bandwidth theorem for locally dense graphs

The Bandwidth theorem of B\"ottcher, Schacht and Taraz gives a condition on the minimum degree of an $n$-vertex graph $G$ that ensures $G$ contains every $r$-chromatic graph $H$ on $n$ vertices of bounded degree and of bandwidth $o(n)$, thereby proving a conjecture of Bollob\'as and Koml\'os. In this paper we prove a version of the Bandwidth theorem for locally dense graphs. Indeed, we prove that every locally dense $n$-vertex graph $G$ with $\delta (G) > (1/2+o(1))n$ contains as a subgraph any given (spanning) $H$ with bounded maximum degree and sublinear bandwidth.Comment: 35 pages. Author accepted version, to appear in Forum of Mathematics, Sigm

Triangle-Tilings in Graphs Without Large Independent Sets

We study the minimum degree necessary to guarantee the existence of perfect and almost-perfect triangle-tilings in an n-vertex graph G with sublinear independence number. In this setting, we show that if δ(G) ≥ n/3 + o(n), then G has a triangle-tiling covering all but at most four vertices. Also, for every r ≥ 5, we asymptotically determine the minimum degree threshold for a perfect triangle-tiling under the additional assumptions that G is Kr-free and n is divisible by 3