41 research outputs found
On directed versions of the Hajnal-Szemerédi theorem
We say that a (di)graph G has a perfect H-packing if there exists a set of vertex-disjoint copies of H which cover all the vertices in G. The seminal HajnalâSzemerĂ©di theorem characterizes the minimum degree that ensures a graph G contains a perfect Kr-packing. In this paper we prove the following analogue for directed graphs: Suppose that T is a tournament on r vertices and G is a digraph of sufficiently large order n where r divides n. If G has minimum in- and outdegree at least (1â1/r)n then G contains a perfect T-packing.In the case when T is a cyclic triangle, this result verifies a recent conjecture of Czygrinow, Kierstead and Molla [4] (for large digraphs). Furthermore, in the case when T is transitive we conjecture that it suffices for every vertex in G to have sufficiently large indegree or outdegree. We prove this conjecture for transitive triangles and asymptotically for all r â©Ÿ 3. Our approach makes use of a result of Keevash and Mycroft [10] concerning almost perfect matchings in hypergraphs as well as the Directed Graph Removal Lemma [1, 6].</jats:p
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)
Crossing Patterns in Nonplanar Road Networks
We define the crossing graph of a given embedded graph (such as a road
network) to be a graph with a vertex for each edge of the embedding, with two
crossing graph vertices adjacent when the corresponding two edges of the
embedding cross each other. In this paper, we study the sparsity properties of
crossing graphs of real-world road networks. We show that, in large road
networks (the Urban Road Network Dataset), the crossing graphs have connected
components that are primarily trees, and that the remaining non-tree components
are typically sparse (technically, that they have bounded degeneracy). We prove
theoretically that when an embedded graph has a sparse crossing graph, it has
other desirable properties that lead to fast algorithms for shortest paths and
other algorithms important in geographic information systems. Notably, these
graphs have polynomial expansion, meaning that they and all their subgraphs
have small separators.Comment: 9 pages, 4 figures. To appear at the 25th ACM SIGSPATIAL
International Conference on Advances in Geographic Information Systems(ACM
SIGSPATIAL 2017
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
Transversal factors and spanning trees
Given a collection of graphs G = (G1,...,Gm) with the same vertex set, an m-edge graph H â âȘiâ[m]Gi is a transversal if there is a bijection Ï : E(H) â [m] such that e â E(GÏ(e)) for each e â E(H). We give asymptotically-tight minimum degree conditions for a graph collection on an n-vertex set to have a transversal which is a copy of a graph H, when H is an n-vertex graph which is an F-factor or a tree with maximum degree o(n/logn)
Products of Differences over Arbitrary Finite Fields
There exists an absolute constant such that for all and all
subsets of the finite field with elements, if
, then Any suffices for sufficiently large
. This improves the condition , due to Bennett, Hart,
Iosevich, Pakianathan, and Rudnev, that is typical for such questions.
Our proof is based on a qualitatively optimal characterisation of sets for which the number of solutions to the equation is nearly
maximum.
A key ingredient is determining exact algebraic structure of sets for
which is nearly minimum, which refines a result of Bourgain and
Glibichuk using work of Gill, Helfgott, and Tao.
We also prove a stronger statement for when are sets in a prime field,
generalising a result of Roche-Newton, Rudnev, Shkredov, and the authors.Comment: 42 page
Fair Interval Scheduling of Indivisible Chores
We study the problem of fairly assigning a set of discrete tasks (or chores)
among a set of agents with additive valuations. Each chore is associated with a
start and finish time, and each agent can perform at most one chore at any
given time. The goal is to find a fair and efficient schedule of the chores,
where fairness pertains to satisfying envy-freeness up to one chore (EF1) and
efficiency pertains to maximality (i.e., no unallocated chore can be feasibly
assigned to any agent). Our main result is a polynomial-time algorithm for
computing an EF1 and maximal schedule for two agents under monotone valuations
when the conflict constraints constitute an arbitrary interval graph. The
algorithm uses a coloring technique in interval graphs that may be of
independent interest. For an arbitrary number of agents, we provide an
algorithm for finding a fair schedule under identical dichotomous valuations
when the constraints constitute a path graph. We also show that stronger
fairness and efficiency properties, including envy-freeness up to any chore
(EFX) along with maximality and EF1 along with Pareto optimality, cannot be
achieved
Multicommodity Multicast, Wireless and Fast
We study rumor spreading in graphs, specifically multicommodity multicast problem under the wireless model: given source-destination pairs in the graph, one needs to find the fastest schedule to transfer information from each source to the corresponding destination. Under the wireless model, nodes can transmit to any subset of their neighbors in synchronous time steps, as long as they either transmit or receive from at most one transmitter during the same time step. We improve approximation ratio for this problem from O~(n^(2/3)) to O~(n^((1/2) + epsilon)) on n-node graphs. We also design an algorithm that satisfies p given demand pairs in O(OPT + p) steps, where OPT is the length of an optimal schedule, by reducing it to the well-studied packet routing problem. In the case where underlying graph is an n-node tree, we improve the previously best-known approximation ratio of O((log n)/(log log n)) to 3. One consequence of our proof is a simple constructive rule for optimal broadcasting in a tree under a widely studied telephone model
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