Hamilton cycles in graphs and hypergraphs: an extremal perspective

As one of the most fundamental and well-known NP-complete problems, the Hamilton cycle problem has been the subject of intensive research. Recent developments in the area have highlighted the crucial role played by the notions of expansion and quasi-randomness. These concepts and other recent techniques have led to the solution of several long-standing problems in the area. New aspects have also emerged, such as resilience, robustness and the study of Hamilton cycles in hypergraphs. We survey these developments and highlight open problems, with an emphasis on extremal and probabilistic approaches.Comment: to appear in the Proceedings of the ICM 2014; due to given page limits, this final version is slightly shorter than the previous arxiv versio

Bounds on the maximum multiplicity of some common geometric graphs

We obtain new lower and upper bounds for the maximum multiplicity of some weighted and, respectively, non-weighted common geometric graphs drawn on n points in the plane in general position (with no three points collinear): perfect matchings, spanning trees, spanning cycles (tours), and triangulations. (i) We present a new lower bound construction for the maximum number of triangulations a set of n points in general position can have. In particular, we show that a generalized double chain formed by two almost convex chains admits {\Omega}(8.65^n) different triangulations. This improves the bound {\Omega}(8.48^n) achieved by the double zig-zag chain configuration studied by Aichholzer et al. (ii) We present a new lower bound of {\Omega}(12.00^n) for the number of non-crossing spanning trees of the double chain composed of two convex chains. The previous bound, {\Omega}(10.42^n), stood unchanged for more than 10 years. (iii) Using a recent upper bound of 30^n for the number of triangulations, due to Sharir and Sheffer, we show that n points in the plane in general position admit at most O(68.62^n) non-crossing spanning cycles. (iv) We derive lower bounds for the number of maximum and minimum weighted geometric graphs (matchings, spanning trees, and tours). We show that the number of shortest non-crossing tours can be exponential in n. Likewise, we show that both the number of longest non-crossing tours and the number of longest non-crossing perfect matchings can be exponential in n. Moreover, we show that there are sets of n points in convex position with an exponential number of longest non-crossing spanning trees. For points in convex position we obtain tight bounds for the number of longest and shortest tours. We give a combinatorial characterization of the longest tours, which leads to an O(nlog n) time algorithm for computing them

Faster exponential-time algorithms in graphs of bounded average degree

We first show that the Traveling Salesman Problem in an n-vertex graph with average degree bounded by d can be solved in O*(2^{(1-\eps_d)n}) time and exponential space for a constant \eps_d depending only on d, where the O*-notation suppresses factors polynomial in the input size. Thus, we generalize the recent results of Bjorklund et al. [TALG 2012] on graphs of bounded degree. Then, we move to the problem of counting perfect matchings in a graph. We first present a simple algorithm for counting perfect matchings in an n-vertex graph in O*(2^{n/2}) time and polynomial space; our algorithm matches the complexity bounds of the algorithm of Bjorklund [SODA 2012], but relies on inclusion-exclusion principle instead of algebraic transformations. Building upon this result, we show that the number of perfect matchings in an n-vertex graph with average degree bounded by d can be computed in O*(2^{(1-\eps_{2d})n/2}) time and exponential space, where \eps_{2d} is the constant obtained by us for the Traveling Salesman Problem in graphs of average degree at most 2d. Moreover we obtain a simple algorithm that counts the number of perfect matchings in an n-vertex bipartite graph of average degree at most d in O*(2^{(1-1/(3.55d))n/2}) time, improving and simplifying the recent result of Izumi and Wadayama [FOCS 2012].Comment: 10 page

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,β)-bijumbled if for any vertex sets A, B ⊆ V (G), we have e( A, B) = p| A||B| ± β√|A||B|. We prove that for any 3 ≤ r ∈ N and c > 0 there exists an ε > 0 such that any n-vertex (p, β)-bijumbled graph with n ∈ rN, δ(G) ≥ c p n and β ≤ ε 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 Szabó 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 β = o( p^2n) actually guarantees that a (p, β)-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 δ(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 Corrádi 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 δ(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 Szemerédi (and Corrádi 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 überdecken. 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,β)-bijumbled, wenn für beliebige Knotenmengen A, B ⊆ V (G) gilt e( A, B) = p| A||B| ± β√|A||B|. Wir beweisen, dass es für jedes 3 ≤ r ∈ N und c > 0 ein ε > 0 gibt, so dass jeder n-Knoten (p, β)-bijumbled Graph mit n ∈ rN, δ(G) ≥ c p n und β ≤ ε p^{r −1} n, einen K_r -Faktor enthält. Dies impliziert ein entsprechendes Ergebnis für den stärkeren Pseudozufallsbegriff von (n, d, λ)-Graphen. Im Fall von K_3-Faktoren, löst dieses Ergebnis eine Vermutung von Krivelevich, Sudakov und Szabó aus dem Jahr 2004 und ist durch eine pseudozufällige K_3-freie Konstruktion von Alon bestmöglich. Tatsächlich ist in diesem Fall noch mehr wahr: als Korollar dieses Ergebnisses können wir schließen, dass die gleiche Bedingung von β = o( p^2n) garantiert, dass ein (p, β)-bijumbled Graph G jeden Graphen mit maximalem Grad 2 enthält. Zweitens untersuchen wir den Begriff der Robustheit für K_3-Faktoren. Für einen Graphen G und p ∈ [0, 1] bezeichnen wir mit G_p die zufällige Sparsifizierung von G, die man erhält, indem man jede Kante von G unabhängig von den anderen Kanten mit einer Wahrscheinlichkeit p behält. Wir zeigen, dass, wenn p ≥ C (log n)^{1/3}n^{−2/3} und G ein n-Knoten-Graph mit n ∈ 3N und δ(G) ≥ 2n/3 ist, G_pmit hoher Wahrscheinlichkeit (mhW) einen K_3-Faktor enthält. Sowohl die Bedingung des minimalen Grades als auch die Wahrscheinlichkeitsbedingung sind bestmöglich. Unser Ergebnis ist eine Verstärkung des klassischen extremalen Satzes von Corrádi und Hajnal, entsprechend p = 1 in unserem Ergebnis, und des berühmten probabilistischen Satzes von Johansson, Kahn und Vu, der den Schwellenwert für 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 für 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 zufällig gestörten Graphen; ein Modell, bei dem man mit einem dichten Graphen beginnt und dann zufällige Kanten hinzufügt. Wir bestimmen, bei gegebenem 0 < α < 1 − 1/r, wie viele zufällige Kanten man zu einem n-Knoten-Graphen G mit δ(G) ≥ α n hinzufügen muss, um sicherzustellen, dass der resultierende Graph mhW einen K_r -Faktor enthält. Wir zeigen, dass, wenn man α erhöht, die Anzahl der benötigten Zufallskanten in regelmäßigen Abständen “springt", und innerhalb dieser Abstände unser Ergebnis bestmöglich ist. Diese Arbeit schließt somit die Lücke zwischen der oben erwähnten bahnbrechenden Arbeit von Johansson, Kahn und Vu, die den rein zufälligen Fall, d.h. α = 0, löst, und der Arbeit von Hajnal und Szemerédi (und Corrádi und Hajnal für r = 3), die zeigt, dass der ursprüngliche Graph bereits den gewünschten K_r -Faktor enthält, wenn α ≥ 1 − 1/r ist