30 research outputs found
Edge-dominating cycles in graphs
AbstractA set S of vertices in a graph G is said to be an edge-dominating set if every edge in G is incident with a vertex in S. A cycle in G is said to be a dominating cycle if its vertex set is an edge-dominating set. Nash-Williams [Edge-disjoint hamiltonian circuits in graphs with vertices of large valency, Studies in Pure Mathematics, Academic Press, London, 1971, pp. 157â183] has proved that every longest cycle in a 2-connected graph of order n and minimum degree at least 13(n+2) is a dominating cycle. In this paper, we prove that for a prescribed positive integer k, under the same minimum degree condition, if n is sufficiently large and if we take k disjoint cycles so that they contain as many vertices as possible, then these cycles form an edge-dominating set. Nash-Williamsâ Theorem corresponds to the case of k=1 of this result
Covering and tiling hypergraphs with tight cycles
Given , we say that a -uniform hypergraph is a
tight cycle on vertices if there is a cyclic ordering of the vertices of
such that every consecutive vertices under this ordering form an
edge. We prove that if and , then every -uniform
hypergraph on vertices with minimum codegree at least has
the property that every vertex is covered by a copy of . Our result is
asymptotically best possible for infinitely many pairs of and , e.g.
when and are coprime.
A perfect -tiling is a spanning collection of vertex-disjoint copies
of . When is divisible by , the problem of determining the
minimum codegree that guarantees a perfect -tiling was solved by a
result of Mycroft. We prove that if and is not divisible
by and divides , then every -uniform hypergraph on vertices
with minimum codegree at least has a perfect
-tiling. Again our result is asymptotically best possible for infinitely
many pairs of and , e.g. when and are coprime with even.Comment: Revised version, accepted for publication in Combin. Probab. Compu
Sufficient degree conditions for graph embeddings
In this dissertation, we focus on the sufficient conditions to guarantee one graph being the subgraph of another. In Chapter 2, we discuss list packing, a modification of the idea of graph packing. This is fitting one graph in the complement of another graph. Sauer and Spencer showed a sufficient bound involving maximum degrees, and this was further explored by Kaul and Kostochka to characterize all extremal cases. Bollobas and Eldridge (and independently Sauer and Spencer) developed edge sum bounds to guarantee packing.
In Chapter 2, we introduce the new idea of list packing and use it to prove stronger versions of many existing theorems. Namely, for two graphs, if the product of the maximum degrees is small or if the total number of edges is small, then the graphs pack.
In Chapter 3, we discuss the problem of finding k vertex-disjoint cycles in a multigraph. This problem originated from a conjecture of Erdos and has led to many different results. Corradi and Hajnal looked at a minimum degree condition. Enomoto and Wang independently looked at a minimum degree-sum condition. More recently, Kierstead, Kostochka, and Yeager characterized the extremal cases to improve these bounds. In Chapter 3, we improve on the multigraph degree-sum result. We characterize all multigraphs that have simple Ore-degree at least 4k -3 , but do not contain k vertex-disjoint cycles. Moreover, we provide a polynomial time algorithm for deciding if a graph contains k vertex-disjoint cycles.
Lastly, in Chapter 4, we consider the same problem but with chorded cycles. Finkel looked at the minimum degree condition while Chiba, Fujita, Gao, and Li addressed the degree-sum condition. More recently, Molla, Santana, and Yeager improved this degree-sum result, and in Chapter 4, we will improve on this further
Tilings in randomly perturbed graphs: Bridging the gap between HajnalâSzemerĂ©di and JohanssonâKahnâVu
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 â 1âr we determine how many random edges one must add to an n-vertex graph G of minimum degree (G) â„ 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 âjumpsâ 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 â„ 1 â 1âr the initial graph already houses the desired perfect Kr-tiling)
Clique Factors: Extremal and Probabilistic Perspectives
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 â V (G), we have e( A, B) = p| A||B| ± ÎČâ|A||B|. We prove that for any 3 †r â N and c > 0 there exists an Δ > 0 such that any n-vertex (p, ÎČ)-bijumbled graph with n â rN, ÎŽ(G) â„ c p n and ÎČ â€ Î” p^{r â1} 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 â [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 â„ C (log n)^{1/3}n^{â2/3} and G is an n-vertex graph with n â 3N and ÎŽ(G) â„ 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 â 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 âjumpsâ 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 â„ 1 â 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 heiĂt (p,ÎČ)-bijumbled, wenn fĂŒr beliebige Knotenmengen A, B â V (G) gilt e( A, B) = p| A||B| ± ÎČâ|A||B|. Wir beweisen, dass es fĂŒr jedes 3 †r â N und c > 0 ein Δ > 0 gibt, so dass jeder n-Knoten (p, ÎČ)-bijumbled Graph mit n â rN, ÎŽ(G) â„ c p n und ÎČ â€ Î” p^{r â1} 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 schlieĂen, 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 â [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 â„ C (log n)^{1/3}n^{â2/3} und G ein n-Knoten-Graph mit n â 3N und ÎŽ(G) â„ 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.
SchlieĂlich 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 â 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€Ăigen AbstĂ€nden âspringt", und innerhalb dieser AbstĂ€nde unser Ergebnis bestmöglich ist. Diese Arbeit schlieĂt 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 â„ 1 â 1/r ist
Extremal problems in graphs
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 contains an -tiling covering an -proportion of the vertices of (for any fixed (0, 1) and graph ). 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 -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 . Specifically, for fixed graphs we study the asymptotic threshold function for the property â . Rödl and RuciĆski [61, 62, 63] determined the threshold function for the general symmetric case; that is, when . 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 Kreuterâs 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)
Extremal graph colouring and tiling problems
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 â 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 â, r, every family F = {F1, F2, . . .} of graphs with v (Fn) = n and â(Fn) †â for every n â 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 + â8)/17 â 0.87226 and further show that this is best possible. This settles a problem of ErdĆs 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 CorrĂĄdiâHajnal theorem and (a special case of) the JohanssonâKahnâVu theorem. We prove that there is some constant C > 0 such that the following is true for every n â 3N and every p â„ Cnâ2/3 (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
On the Existence of Loose Cycle Tilings and Rainbow Cycles
abstract: Extremal graph theory results often provide minimum degree
conditions which guarantee a copy of one graph exists within
another. A perfect -tiling of a graph is a collection
of subgraphs of such that every element of
is isomorphic to and such that every vertex in
is in exactly one element of . Let denote
the loose cycle on vertices, the -uniform hypergraph
obtained by replacing the edges of a graph cycle
on vertices with edge triples , where is
uniquely assigned to . This dissertation proves for even
, that any sufficiently large -uniform hypergraph
on vertices with minimum -degree
\delta^1(H) \geq {n - 1 \choose 2} - {\Bsize \choose 2} + c(t,n) +
1, where , contains a perfect
-tiling. The result is tight, generalizing previous
results on by Han and Zhao. For an edge colored graph ,
let the minimum color degree be the minimum number of
distinctly colored edges incident to a vertex. Call rainbow if
every edge has a unique color. For , this dissertation
proves that any sufficiently large edge colored graph on
vertices with contains a rainbow
cycle on vertices. The result is tight for odd and
extends previous results for . In addition, for even
, this dissertation proves that any sufficiently large
edge colored graph on vertices with
, where
, contains a rainbow cycle on
vertices. The result is tight when . As a related
result, this dissertation proves for all , that any
sufficiently large oriented graph on vertices with
contains a directed cycle on
vertices. This partially generalizes a result by Kelly,
K\"uhn, and Osthus that uses minimum semidegree rather than minimum
out degree.Dissertation/ThesisDoctoral Dissertation Mathematics 201
Disjoint isomorphic balanced clique subdivisions
A classical result, due to Bollobås and Thomason, and independently Komlós and Szemerédi, states that there is a constant C such that every graph with average degree at least has a subdivision of , the complete graph on k vertices. We study two directions extending this result. ⹠Verstraëte conjectured that a quadratic bound guarantees in fact two vertex-disjoint isomorphic copies of a -subdivision.
⹠Thomassen conjectured that for each there is some such that every graph with average degree at least d contains a balanced subdivision of . Recently, Liu and Montgomery confirmed Thomassen's conjecture, but the optimal bound on remains open. In this paper, we show that a quadratic lower bound on average degree suffices to force a balanced -subdivision. This gives the right order of magnitude of the optimal needed in Thomassen's conjecture. Since a balanced -subdivision trivially contains m vertex-disjoint isomorphic -subdivisions, this also confirms Verstraëte's conjecture in a strong sense
The Extremal Function for Two Disjoint Cycles *
Abstract A theta graph is the union of three internally disjoint paths that have the same two distinct end vertices. We show that every graph of order n â„ 9 and size at least 7nâ13 2 contains two disjoint theta graphs. We also show that every 2-edge-connected graph of order n â„ 6 and size at least 3n â 5 contains two disjoint cycles, such that any specified vertex with degree at least three belongs to one of them. The lower bound on size in both are sharp in general