The two largest distances in finite planar sets

AbstractWe determine all homogenous linear inequalities satisfied by the numbers of occurrences of the two largest distances among n points in the plane

Nondeterministic graph property testing

A property of finite graphs is called nondeterministically testable if it has a "certificate" such that once the certificate is specified, its correctness can be verified by random local testing. In this paper we study certificates that consist of one or more unary and/or binary relations on the nodes, in the case of dense graphs. Using the theory of graph limits, we prove that nondeterministically testable properties are also deterministically testable.Comment: Version 2: 11 pages; we allow orientation in the certificate, describe new application

Bracing for a no-deal brexit

We define a distance of two graphs that reflects the closeness of both local and global properties, We also define convergence of a sequence of graphs, and show that a graph sequence is convergent if and only if it is Cauchy in this distance. Every convergent graph sequence has a limit in the form of a symmetric measurable function in two variables. We use these notions of distance and graph limits to give a general theory for parameter testing. As examples, we provide short proofs of the testability of MaxCut and the recent result of Alon and Shapira about the testability of hereditary graph properties. Copyright 2006 ACM

Diszkrét és folytonos: a gráfelmélet, algebra, analízis és geometria találkozási pontjai = Discrete and Continuous: interfaces between graph theory, algebra, analysis and geometry

Sok eredmény született a gráfok növekvő konvergens sorozataival és azok limesz-objektumaival, ill. az ezek vizsgálatára szolgáló gráf-algebrákkal kapcsolatban. Kidolgozásra kerültek a nagyon nagy sűrű gráfok (hálózatok) matematikai elméletének alapjai, és ezek alkalmazásai az extremális gráfelmélet területén. Aktív és eredményes kutatás folyt a diszkrét matematika más, klasszikus matematikai területekkel való kapcsolatával kapcsolatban: topológia (a topológiai módszer alkalmazása gráfok magjára, ill a csomók elmélete), geometriai szerkezetek merevsége (a Molekuláris Sejtés bizonyítása 2 dimenzióban), diszkrét geometriai (Bang sejtésének bizonyítása), véges geometriák (lefogási problémák, extremális problémák q-analogonjai), algebra (félcsoport varietások, gráfhatványok színezése), számelmélet (additív számelmélet, Heilbronn probléma), továbbá gráfalgoritmusok (stabilis párosítások, biológiai alkalmazások)) területén. | Several results were obtained in connection with convergent growing sequences of graphs and their limit objects, and with graph algebras facilitating their study. Basic concepts for the study of very large dense graphs were worked out, along with their applications to extremal graph theory. Active and successful research was conducted concerning the interaction of discrete mathematics with other, classical areas of mathematics: topology (applications of topology in the study of kernels of graphs, and the theory of knots), rigidity of geometric structures (proof of the Molecular Conjecture in 2 dimensions), discrete geometry (proof of the conjecture of Bang), finite geometries (blocking problems, q-analogues of extremal problems), algebra (semigroup varieties, coloring of graph powers), number theory (additive number theory, heilbronn problem), and graph algorithms (stable matchings, applications in biology)

Convex Quadrilaterals and k-Sets

Introduction Let S be a set of n points in general position in the plane, i.e. no three points are collinear. Four points in S may or may not form the vertices of a convex quadrilateral; if they do, we call this subset of 4 elements convex. We are interested in the number of convex 4-element subsets. This can of course be as large as , if S is in convex position, but what is its minimum? Another way of stating the problem is to find the rectilinear crossing number of the complete n-graph K n , i.e., to determine the minimum number of crossings in a drawing of K n in the plane with straight edges and the nodes in general position. We note here that the rectilinear crossing number of complete graphs also determines the rectilinear crossing number of random graphs (provided the probability for an edge to appear is at least ln n ), as was shown by Spencer and Toth [13]. It is easy to see that for n = 5 we get at least one convex 4-element subset, from which it follows by straig

Counting graph homomorphisms

Counting homomorphisms between graphs (often with weights) comes up in a wide variety of areas, including extremal graph theory, properties of graph products, partition functions in statistical physics and property testing of large graphs. In this paper we survey recent developments in the study of homomorphism numbers, including the characterization of the homomorphism numbers in terms of the semidefiniteness of “connection matrices”, and some applications of this fact in extremal graph theory. We define a distance of two graphs in terms of similarity of their global structure, which also reflects the closeness of (appropriately scaled) homomorphism numbers into the two graphs. We use homomorphism numbers to define convergence of a sequence of graphs, and show that a graph sequence is convergent if and only if it is Cauchy in this distance. Every convergent graph sequence has a limit in the form of a symmetric measurable function in two variables. We use these notions of distance and graph limits to give a general theory for parameter testing. The convergence can also be characterized in terms of mappings of the graphs into fixed small graphs, which is strongly connected to important parameters like ground state energy in statistical physics, and to weighted maximum cut problems in computer science

Limits of randomly grown graph sequences

Motivated in part by various sequences of graphs growing under random rules (such as Internet models), Borgs, Chayes, Lovász, Sós, Szegedy and Vesztergombi introduced convergent sequences of dense graphs and their limits. In this paper we use this framework to study one of the motivating classes of examples, namely randomly growing graphs. We prove the (almost sure) convergence of several such randomly growing graph sequences, and determine their limit. The analysis is not always straightforward: in some cases the cut-distance from a limit object can be directly estimated, while in other cases densities of subgraphs can be shown to converge. © 2011 Elsevier Ltd