19 research outputs found

    Triangles in randomly perturbed graphs

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    We study the problem of finding pairwise vertex-disjoint triangles in the randomly perturbed graph model, which is the union of any n -vertex graph G satisfying a given minimum degree condition and the binomial random graph G(n,p) . We prove that asymptotically almost surely GâˆȘG(n,p) contains at least min{ÎŽ(G),⌊n/3⌋} pairwise vertex-disjoint triangles, provided p≄Clogn/n , where C is a large enough constant. This is a perturbed version of an old result of Dirac. Our result is asymptotically optimal and answers a question of Han, Morris, and Treglown [RSA, 2021, no. 3, 480–516] in a strong form. We also prove a stability version of our result, which in the case of pairwise vertex-disjoint triangles extends a result of Han, Morris, and Treglown [RSA, 2021, no. 3, 480–516]. Together with a result of Balogh, Treglown, and Wagner [CPC, 2019, no. 2, 159–176], this fully resolves the existence of triangle factors in randomly perturbed graphs. We believe that the methods introduced in this paper are useful for a variety of related problems: we discuss possible generalisations to clique factors, cycle factors, and 2 -universality

    Properly colored subgraphs in edge-colored graphs

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    Properly colored and rainbow cycles in edge-colored graphs

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    Proceedings of the Fourth Russian Finnish Symposium on Discrete Mathematics

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    Clique Factors: Extremal and Probabilistic Perspectives

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    A K_r-factor in a graph G is a collection of vertex-disjoint copies of K_r covering the vertex set of G. In this thesis, we investigate these fundamental objects in three settings that lie at the intersection of extremal and probabilistic combinatorics. Firstly, we explore pseudorandom graphs. An n-vertex graph is said to be (p,ÎČ)-bijumbled if for any vertex sets A, B ⊆ V (G), we have e( A, B) = p| A||B| ± ÎČ√|A||B|. We prove that for any 3 ≀ r ∈ N and c > 0 there exists an Δ > 0 such that any n-vertex (p, ÎČ)-bijumbled graph with n ∈ rN, ÎŽ(G) ≄ c p n and ÎČ â‰€ Δ p^{r −1} n, contains a K_r -factor. This implies a corresponding result for the stronger pseudorandom notion of (n, d, λ)-graphs. For the case of K_3-factors, this result resolves a conjecture of Krivelevich, Sudakov and SzabĂł from 2004 and it is tight due to a pseudorandom triangle-free construction of Alon. In fact, in this case even more is true: as a corollary to this result, we can conclude that the same condition of ÎČ = o( p^2n) actually guarantees that a (p, ÎČ)-bijumbled graph G contains every graph on n vertices with maximum degree at most 2. Secondly, we explore the notion of robustness for K_3-factors. For a graph G and p ∈ [0, 1], we denote by G_p the random sparsification of G obtained by keeping each edge of G independently, with probability p. We show that there exists a C > 0 such that if p ≄ C (log n)^{1/3}n^{−2/3} and G is an n-vertex graph with n ∈ 3N and ÎŽ(G) ≄ 2n/3 , then with high probability G_p contains a K_3-factor. Both the minimum degree condition and the probability condition, up to the choice of C, are tight. Our result can be viewed as a common strengthening of the classical extremal theorem of CorrĂĄdi and Hajnal, corresponding to p = 1 in our result, and the famous probabilistic theorem of Johansson, Kahn and Vu establishing the threshold for the appearance of K_3-factors (and indeed all K_r -factors) in G (n, p), corresponding to G = K_n in our result. It also implies a first lower bound on the number of K_3-factors in graphs with minimum degree at least 2n/3, which gets close to the truth. Lastly, we consider the setting of randomly perturbed graphs; a model introduced by Bohman, Frieze and Martin, where one starts with a dense graph and then adds random edges to it. Specifically, given any fixed 0 < α < 1 − 1/r we determine how many random edges one must add to an n-vertex graph G with ÎŽ(G) ≄ α n to ensure that, with high probability, the resulting graph contains a K_r -factor. As one increases α we demonstrate that the number of random edges required ‘jumps’ at regular intervals, and within these intervals our result is best-possible. This work therefore bridges the gap between the seminal work of Johansson, Kahn and Vu mentioned above, which resolves the purely random case, i.e., α = 0, and that of Hajnal and SzemerĂ©di (and CorrĂĄdi and Hajnal for r = 3) showing that when α ≄ 1 − 1/r the initial graph already hosts the desired K_r -factor.Ein K_r -Faktor in einem Graphen G ist eine Sammlung von Knoten-disjunkten Kopien von K_r , die die Knotenmenge von G ĂŒberdecken. Wir untersuchen diese Objekte in drei Kontexten, die an der Schnittstelle zwischen extremaler und probabilistischer Kombinatorik liegen. Zuerst untersuchen wir Pseudozufallsgraphen. Ein Graph heißt (p,ÎČ)-bijumbled, wenn fĂŒr beliebige Knotenmengen A, B ⊆ V (G) gilt e( A, B) = p| A||B| ± ÎČ√|A||B|. Wir beweisen, dass es fĂŒr jedes 3 ≀ r ∈ N und c > 0 ein Δ > 0 gibt, so dass jeder n-Knoten (p, ÎČ)-bijumbled Graph mit n ∈ rN, ÎŽ(G) ≄ c p n und ÎČ â‰€ Δ p^{r −1} n, einen K_r -Faktor enthĂ€lt. Dies impliziert ein entsprechendes Ergebnis fĂŒr den stĂ€rkeren Pseudozufallsbegriff von (n, d, λ)-Graphen. Im Fall von K_3-Faktoren, löst dieses Ergebnis eine Vermutung von Krivelevich, Sudakov und SzabĂł aus dem Jahr 2004 und ist durch eine pseudozufĂ€llige K_3-freie Konstruktion von Alon bestmöglich. TatsĂ€chlich ist in diesem Fall noch mehr wahr: als Korollar dieses Ergebnisses können wir schließen, dass die gleiche Bedingung von ÎČ = o( p^2n) garantiert, dass ein (p, ÎČ)-bijumbled Graph G jeden Graphen mit maximalem Grad 2 enthĂ€lt. Zweitens untersuchen wir den Begriff der Robustheit fĂŒr K_3-Faktoren. FĂŒr einen Graphen G und p ∈ [0, 1] bezeichnen wir mit G_p die zufĂ€llige Sparsifizierung von G, die man erhĂ€lt, indem man jede Kante von G unabhĂ€ngig von den anderen Kanten mit einer Wahrscheinlichkeit p behĂ€lt. Wir zeigen, dass, wenn p ≄ C (log n)^{1/3}n^{−2/3} und G ein n-Knoten-Graph mit n ∈ 3N und ÎŽ(G) ≄ 2n/3 ist, G_pmit hoher Wahrscheinlichkeit (mhW) einen K_3-Faktor enthĂ€lt. Sowohl die Bedingung des minimalen Grades als auch die Wahrscheinlichkeitsbedingung sind bestmöglich. Unser Ergebnis ist eine VerstĂ€rkung des klassischen extremalen Satzes von CorrĂĄdi und Hajnal, entsprechend p = 1 in unserem Ergebnis, und des berĂŒhmten probabilistischen Satzes von Johansson, Kahn und Vu, der den Schwellenwert fĂŒr das Auftreten eines K_3-Faktors (und aller K_r -Faktoren) in G (n, p) festlegt, entsprechend G = K_n in unserem Ergebnis. Es impliziert auch eine erste untere Schranke fĂŒr die Anzahl der K_3-Faktoren in Graphen mit einem minimalen Grad von mindestens 2n/3, die der Wahrheit nahe kommt. Schließlich betrachten wir die Situation von zufĂ€llig gestörten Graphen; ein Modell, bei dem man mit einem dichten Graphen beginnt und dann zufĂ€llige Kanten hinzufĂŒgt. Wir bestimmen, bei gegebenem 0 < α < 1 − 1/r, wie viele zufĂ€llige Kanten man zu einem n-Knoten-Graphen G mit ÎŽ(G) ≄ α n hinzufĂŒgen muss, um sicherzustellen, dass der resultierende Graph mhW einen K_r -Faktor enthĂ€lt. Wir zeigen, dass, wenn man α erhöht, die Anzahl der benötigten Zufallskanten in regelmĂ€ĂŸigen AbstĂ€nden “springt", und innerhalb dieser AbstĂ€nde unser Ergebnis bestmöglich ist. Diese Arbeit schließt somit die LĂŒcke zwischen der oben erwĂ€hnten bahnbrechenden Arbeit von Johansson, Kahn und Vu, die den rein zufĂ€lligen Fall, d.h. α = 0, löst, und der Arbeit von Hajnal und SzemerĂ©di (und CorrĂĄdi und Hajnal fĂŒr r = 3), die zeigt, dass der ursprĂŒngliche Graph bereits den gewĂŒnschten K_r -Faktor enthĂ€lt, wenn α ≄ 1 − 1/r ist

    Proceedings of the Fourth Russian Finnish Symposium on Discrete Mathematics

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    On some problems in extremal hypergraph theory

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    In this thesis, we study several problems in extremal (hyper)graph theory. We begin by investigating the problem of subgraph containment in the model of randomly perturbed graphs. In particular, we study the perturbed threshold for the appearance of the square of a Hamilton cycle and the problem of finding pairwise vertex-disjoint triangles. We provide a stability version of these results and we discuss their implications on the perturbed thresholds for 2-universality and for a triangle factor. We then turn to the notion of threshold in the context of transversals in hypergraph collections. Here the question is, given a fixed m-edge hypergraph F, how large the minimum degree of each hypergraph Hi needs to be, so that the hypergraph collection (H1, 
, Hm) necessarily contains a transversal copy of F. We prove a widely applicable sufficient condition on F such that the following holds. The needed minimum degree is asymptotically the same as the minimum degree required for a copy of F to appear in each Hi. The condition is general enough to obtain transversal variants of various classical Dirac-type results for (powers of) Hamilton cycles. Finally, we initiate the study of a new variant of the Maker-Breaker positional game, which we call the (1:b) multistage Maker-Breaker game. Starting with a given hypergraph, we play several stages of a usual (1:b) Maker-Breaker game where, in each stage, we shrink the board by keeping only the elements that Maker claimed in the previous stage and updating the collection of winning sets accordingly. The game proceeds until no winning sets remain, and the goal of Maker is to prolong the duration of the game for as many stages as possible. We estimate the maximum duration of the (1:b) multistage Maker-Breaker game for several standard graph games played on the edge set of Kn with biases b subpolynomial in n

    Extremal problems in graphs

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    In the first part of this thesis we will consider degree sequence results for graphs. An important result of KomlĂłs [39] yields the asymptotically exact minimum degree threshold that ensures a graph GG contains an HH-tiling covering an xx-proportion of the vertices of GG (for any fixed x∈x ∈ (0, 1) and graph HH). In Chapter 2, we give a degree sequence strengthening of this result. A fundamental result of KĂŒhn and Osthus [46] determines up to an additive constant the minimum degree threshold that forces a graph to contain a perfect HH-tiling. In Chapter 3, we prove a degree sequence version of this result. We close this thesis in the study of asymmetric Ramsey properties in Gn,pG_n,_p. Specifically, for fixed graphs H1,...,Hr,H_1, . . . , H_r, we study the asymptotic threshold function for the property Gn,pG_n,_p → H1,...,HrH_1, . . . , H_r. Rödl and RuciƄski [61, 62, 63] determined the threshold function for the general symmetric case; that is, when H1=⋅⋅⋅=HrH_1 = · · · = H_r. Kohayakawa and Kreuter [33] conjectured the threshold function for the asymmetric case. Building on work of Marciniszyn, Skokan, Spöhel and Steger [51], in Chapter 4, we reduce the 0-statement of Kohayakawa and Kreuter’s conjecture to a more approachable, deterministic conjecture. To demonstrate the potential of this approach, we show our conjecture holds for almost all pairs of regular graphs (satisfying certain balancedness conditions)

    Tilings in randomly perturbed graphs: Bridging the gap between Hajnal‐SzemerĂ©di and Johansson‐Kahn‐Vu

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    A perfect Kr-tiling in a graph G is a collection of vertex-disjoint copies of Kr that together cover all the vertices in G. In this paper we consider perfect Kr-tilings in the setting of randomly perturbed graphs; a model introduced by Bohman, Frieze, and Martin [7] where one starts with a dense graph and then adds m random edges to it. Specifically, given any fixed 0 < < 1 − 1∕r we determine how many random edges one must add to an n-vertex graph G of minimum degree (G) ≄ n to ensure that, asymptotically almost surely, the resulting graph contains a perfect Kr-tiling. As one increases we demonstrate that the number of random edges required “jumps” at regular intervals, and within these intervals our result is best-possible. This work therefore closes the gap between the seminal work of Johansson, Kahn and Vu [25] (which resolves the purely random case, that is, = 0) and that of Hajnal and SzemerĂ©di [18] (which demonstrates that for ≄ 1 − 1∕r the initial graph already houses the desired perfect Kr-tiling)

    Extremal graph colouring and tiling problems

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    In this thesis, we study a variety of different extremal graph colouring and tiling problems in finite and infinite graphs. Confirming a conjecture of GyĂĄrfĂĄs, we show that for all k, r ∈ N there is a constant C > 0 such that the vertices of every r-edge-coloured complete k-uniform hypergraph can be partitioned into a collection of at most C monochromatic tight cycles. We shall say that the family of tight cycles has finite r-colour tiling number. We further prove that, for all natural numbers k, p and r, the family of p-th powers of k-uniform tight cycles has finite r-colour tiling number. The case where k = 2 settles a problem of Elekes, Soukup, Soukup and SzentmiklĂłssy. We then show that for all natural numbers ∆, r, every family F = {F1, F2, . . .} of graphs with v (Fn) = n and ∆(Fn) ≀ ∆ for every n ∈ N has finite r-colour tiling number. This makes progress on a conjecture of Grinshpun and SĂĄrközy. We study Ramsey problems for infinite graphs and prove that in every 2-edge- colouring of KN, the countably infinite complete graph, there exists a monochromatic infinite path P such that V (P) has upper density at least (12 + √8)/17 ≈ 0.87226 and further show that this is best possible. This settles a problem of ErdƑs and Galvin. We study similar problems for many other graphs including trees and graphs of bounded degree or degeneracy and prove analogues of many results concerning graphs with linear Ramsey number in finite Ramsey theory. We also study a different sort of tiling problem which combines classical problems from extremal and probabilistic graph theory, the CorrĂĄdi–Hajnal theorem and (a special case of) the Johansson–Kahn–Vu theorem. We prove that there is some constant C > 0 such that the following is true for every n ∈ 3N and every p ≄ Cn−2/3 (log n)1/3. If G is a graph on n vertices with minimum degree at least 2n/3, then Gp (the random subgraph of G obtained by keeping every edge independently with probability p) contains a triangle tiling with high probability
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