2,274 research outputs found
Monochromatic Clique Decompositions of Graphs
Let be a graph whose edges are coloured with colours, and be a -tuple of graphs. A monochromatic -decomposition of is a partition of the edge set of such that each
part is either a single edge or forms a monochromatic copy of in colour
, for some . Let be the smallest
number , such that, for every order- graph and every
-edge-colouring, there is a monochromatic -decomposition with at
most elements. Extending the previous results of Liu and Sousa
["Monochromatic -decompositions of graphs", Journal of Graph Theory},
76:89--100, 2014], we solve this problem when each graph in is a
clique and is sufficiently large.Comment: 14 pages; to appear in J Graph Theor
A Semidefinite Approach to the Cover Problem
We apply theta body relaxations to the -cover problem and show
polynomial time solvability for certain classes of graphs. In particular, we
give an effective relaxation where all --hole facets are valid, and
study its relation to an open question of Conforti et al. For the triangle free
problem, we show for that the theta body relaxations do not converge by
steps; we also prove for all an integrality gap of 2 for the
second theta body
Hypergraph matchings and designs
We survey some aspects of the perfect matching problem in hypergraphs, with
particular emphasis on structural characterisation of the existence problem in
dense hypergraphs and the existence of designs.Comment: 19 pages, for the 2018 IC
A Geometric Theory for Hypergraph Matching
We develop a theory for the existence of perfect matchings in hypergraphs
under quite general conditions. Informally speaking, the obstructions to
perfect matchings are geometric, and are of two distinct types: 'space
barriers' from convex geometry, and 'divisibility barriers' from arithmetic
lattice-based constructions. To formulate precise results, we introduce the
setting of simplicial complexes with minimum degree sequences, which is a
generalisation of the usual minimum degree condition. We determine the
essentially best possible minimum degree sequence for finding an almost perfect
matching. Furthermore, our main result establishes the stability property:
under the same degree assumption, if there is no perfect matching then there
must be a space or divisibility barrier. This allows the use of the stability
method in proving exact results. Besides recovering previous results, we apply
our theory to the solution of two open problems on hypergraph packings: the
minimum degree threshold for packing tetrahedra in 3-graphs, and Fischer's
conjecture on a multipartite form of the Hajnal-Szemer\'edi Theorem. Here we
prove the exact result for tetrahedra and the asymptotic result for Fischer's
conjecture; since the exact result for the latter is technical we defer it to a
subsequent paper.Comment: Accepted for publication in Memoirs of the American Mathematical
Society. 101 pages. v2: minor changes including some additional diagrams and
passages of expository tex
Substructures in Latin squares
We prove several results about substructures in Latin squares. First, we
explain how to adapt our recent work on high-girth Steiner triple systems to
the setting of Latin squares, resolving a conjecture of Linial that there exist
Latin squares with arbitrarily high girth. As a consequence, we see that the
number of order- Latin squares with no intercalate (i.e., no
Latin subsquare) is at least . Equivalently,
, where is the number
of intercalates in a uniformly random order- Latin square.
In fact, extending recent work of Kwan, Sah, and Sawhney, we resolve the
general large-deviation problem for intercalates in random Latin squares, up to
constant factors in the exponent: for any constant we have
and for
any constant we have
.
Finally, we show that in almost all order- Latin squares, the number of
cuboctahedra (i.e., the number of pairs of possibly degenerate
subsquares with the same arrangement of symbols) is of order , which is
the minimum possible. As observed by Gowers and Long, this number can be
interpreted as measuring "how associative" the quasigroup associated with the
Latin square is.Comment: 32 pages, 1 figur
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
Topological properties and fractal analysis of recurrence network constructed from fractional Brownian motions
Many studies have shown that we can gain additional information on time
series by investigating their accompanying complex networks. In this work, we
investigate the fundamental topological and fractal properties of recurrence
networks constructed from fractional Brownian motions (FBMs). First, our
results indicate that the constructed recurrence networks have exponential
degree distributions; the relationship between and of recurrence networks decreases with the Hurst
index of the associated FBMs, and their dependence approximately satisfies
the linear formula . Moreover, our numerical results of
multifractal analysis show that the multifractality exists in these recurrence
networks, and the multifractality of these networks becomes stronger at first
and then weaker when the Hurst index of the associated time series becomes
larger from 0.4 to 0.95. In particular, the recurrence network with the Hurst
index possess the strongest multifractality. In addition, the
dependence relationships of the average information dimension on the Hurst index can also be
fitted well with linear functions. Our results strongly suggest that the
recurrence network inherits the basic characteristic and the fractal nature of
the associated FBM series.Comment: 25 pages, 1 table, 15 figures. accepted by Phys. Rev.
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