97 research outputs found
Extensions of Extremal Graph Theory to Grids
We consider extensions of Turán\u27s original theorem of 1941 to planar grids. For a complete kxm array of vertices, we establish in Proposition 4.3 an exact formula for the maximal number of edges possible without any square regions. We establish with Theorem 4.12 an upper bound and with Theorem 4.15 an asymptotic lower bound for the maximal number of edges on a general grid graph with n vertices and no rectangles
On Some Applications of Graph Theory, I
In a series of papers, of which the present one is Part 1, it is shown that solutions to a variety of problems in distance geometry, potential theory and theory of metric spaces are provided by appropriate applications of graph theoretic results. (c) 1972 Published by Elsevier B.V
On the equidistribution properties of patterns in prime numbers Jumping Champions, metaanalysis of properties as Low-Discrepancy Sequences, and some conjectures based on Ramanujan's master theorem and the zeros of Riemann's zeta function
The Paul Erd\H{o}s-Tur\'an inequality is used as a quantitative form of Weyl'
s criterion, together with other criteria to asses equidistribution properties
on some patterns of sequences that arise from indexation of prime numbers,
Jumping Champions (called here and in previous work, "meta-distances" or even
md, for short). A statistical meta-analysis is also made of previous research
concerning meta-distances to review the conclusion that meta-distances can be
called Low-discrepancy sequences (LDS), and thus exhibiting another numerical
evidence that md's are an equidistributed sequence. Ramanujan's master theorem
is used to conjecture that the types of integrands where md's can be used more
succesfully for quadratures are product-related, as opposite to
addition-related. Finally, it is conjectured that the equidistribution of md's
may be connected to the know equidistribution of zeros of Riemann's zeta
function, and yet still have enough "information" for quasi-random integration
("right" amount of entropy).Comment: 13 pages, 7 Figures, 4 table
Universal and unavoidable graphs
The Tur\'an number of a graph is the maximal number of
edges in an -free graph on vertices. In Chung and Erd\H{o}s asked
which graphs with edges minimize . They resolved this
question asymptotically for most of the range of and asked to complete the
picture. In this paper we answer their question by resolving all remaining
cases. Our result translates directly to the setting of universality, a
well-studied notion of finding graphs which contain every graph belonging to a
certain family. In this setting we extend previous work done by Babai, Chung,
Erd\H{o}s, Graham and Spencer, and by Alon and Asodi.Comment: 14 pages, 4 figure
Closed expressions for averages of set partition statistics
In studying the enumerative theory of super characters' of the group of upper
triangular matrices over a finite field we found that the moments (mean,
variance and higher moments) of novel statistics on set partitions have simple
closed expressions as linear combinations of shifted bell numbers. It is shown
here that families of other statistics have similar moments. The coefficients
in the linear combinations are polynomials in . This allows exact
enumeration of the moments for small to determine exact formulae for all
Turán problems in graphs and hypergraphs
Mantel's theorem says that among all triangle-free graphs of a given order the balanced complete bipartite graph is the unique graph of maximum size. In Chapter 2, we prove an analogue of this result for 3-graphs (3-uniform hy¬pergraphs) together with an associated stability result. Let K− 4 , F5 and F6 be 3-graphs with vertex sets {1, 2,3, 4}, {1, 2,3,4, 5} and {1, 2,3,4, 5, 6} re¬spectively and edge sets E(K−4 ) = {123, 124, 134}, E(F5) = {123, 124, 345}, E(F6) = {123, 124,345, 156} and F = {K4, F6}. For n =6 5 the unique F-free 3-graph of order n and maximum size is the balanced complete tri¬partite 3-graph S3(n). This extends an old result of Bollobas that S3(n) is the unique 3-graph of maximum size with no copy of K− 4 or F5.
In 1941, Turán generalised Mantel's theorem to cliques of arbitrary size and then asked whether similar results could be obtained for cliques on hyper-graphs. This has become one of the central unsolved problems in the field of extremal combinatorics. In Chapter 3, we prove that the Turán density
of K(3)
5 together with six other induced subgraphs is 3/4. This is analogous to a similar result obtained for K(3)
4 by Razborov.
In Chapter 4, we consider various generalisations of the Turán density. For
example, we prove that, if the density in C of ¯P3 is x and C is K3-free, then
|E(C)| /(n ) ≤ 1/4+(1/4)J1 − (8/3)x. This is motivated by the observation
2
that the extremal graph for K3 is ¯P3-free, so that the upper bound is a natural extension of a stability result for K3.
The question how many edges can be deleted from a blow-up of H before it is H-free subject to the constraint that the same proportion of edges are deleted from each connected pair of vertex sets has become known as the Turán density problem. In Chapter 5, using entropy compression supplemented with some analytic methods, we derive an upper bound of 1 − 1/('y(Δ(H) − /3)), where Δ(H) is the maximum degree of H, 3 ≤ 'y < 4 and /3 ≤ 1. The new bound asymptotically approaches the existing best upper bound despite being derived in a completely different way.
The techniques used in these results, illustrating their breadth and connec¬tions between them, are set out in Chapter 1
Provably and Efficiently Approximating Near-cliques using the Tur\'an Shadow: PEANUTS
Clique and near-clique counts are important graph properties with
applications in graph generation, graph modeling, graph analytics, community
detection among others. They are the archetypal examples of dense subgraphs.
While there are several different definitions of near-cliques, most of them
share the attribute that they are cliques that are missing a small number of
edges. Clique counting is itself considered a challenging problem. Counting
near-cliques is significantly harder more so since the search space for
near-cliques is orders of magnitude larger than that of cliques.
We give a formulation of a near-clique as a clique that is missing a constant
number of edges. We exploit the fact that a near-clique contains a smaller
clique, and use techniques for clique sampling to count near-cliques. This
method allows us to count near-cliques with 1 or 2 missing edges, in graphs
with tens of millions of edges. To the best of our knowledge, there was no
known efficient method for this problem, and we obtain a 10x - 100x speedup
over existing algorithms for counting near-cliques.
Our main technique is a space-efficient adaptation of the Tur\'an Shadow
sampling approach, recently introduced by Jain and Seshadhri (WWW 2017). This
approach constructs a large recursion tree (called the Tur\'an Shadow) that
represents cliques in a graph. We design a novel algorithm that builds an
estimator for near-cliques, using an online, compact construction of the
Tur\'an Shadow.Comment: The Web Conference, 2020 (WWW
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