An extension of the Erdős-Szekeres theorem on large angles

The existence of a functionn(ε) (ε>0) is established such that given a finite setV in the plane there exists a subsetW⊆V, |W|<n(ε) with the property that for anyv εV\ W there are two pointsw ₁,w ₂ εW such that the angle ∢(w ₁ vw ₂)>π-ε.     

Properties of complete sets of mutually equiorthogonal frequency hypercubes

A complete set of mutually equiorthogonal frequency hypercubes (MEFH) of ordern and dimensiond, usingm distinct symbols, has (n−1) ^d /(m−1) hypercubes. In this article, we explore the properties of complete sets of MEFH. As a consequence of these properties, we show that existence of such a set implies that the number of symbolsm is a prime power. We also establish an equivalence between existence of a complete set of MEFH and existence of a certain complete set of Latin hypercubes and a certain complete orthogonal array.     

A Complete Solution to Spectrum Problem for Five‐Vertex Graphs with Application to Traffic Grooming in Optical Networks

A G‐design of order n is a decomposition of the complete graph on n vertices into edge‐disjoint subgraphs isomorphic to G. Grooming uniform all‐to‐all traffic in optical ring networks with grooming ratio C requires the determination of graph decompositions of the complete graph on n vertices into subgraphs each having at most C edges. The drop cost of such a grooming is the total number of vertices of nonzero degree in these subgraphs, and the grooming is optimal when the drop cost is minimum. The existence spectrum problem of G‐designs for five‐vertex graphs is a long standing problem posed by Bermond, Huang, Rosa and Sotteau in 1980, which is closely related to traffic groomings in optical networks. Although considerable progress has been made over the past 30 years, the existence problems for such G‐designs and their related traffic groomings in optical networks are far from complete. In this paper, we first give a complete solution to this spectrum problem for five‐vertex graphs by eliminating all the undetermined possible exceptions. Then, we determine almost completely the minimum drop cost of 8‐groomings for all orders n by reducing the 37 possible exceptions to 8. Finally, we show the minimum possible drop cost of 9‐groomings for all orders n is realizable with 14 exceptions and 12 possible exceptions.     

The Diameter of a Scale-Free Random Graph

We consider a random graph process in which vertices are added to the graph one at a time and joined to a fixed number m of earlier vertices, where each earlier vertex is chosen with probability proportional to its degree. This process was introduced by Barabási and Albert [3], as a simple model of the growth of real-world graphs such as the world-wide web. Computer experiments presented by Barabási, Albert and Jeong [1,5] and heuristic arguments given by Newman, Strogatz and Watts [23] suggest that after n steps the resulting graph should have diameter approximately logn. We show that while this holds for m=1, for m2 the diameter is asymptotically log n/log logn.* Research supported in part by NSF grant no. DSM9971788     

Minimum cutsets for an element of a boolean lattice

An informative new proof is given for the theorem of Nowakowski that determines for all n and k the minimum size of a cutset for an element A with |A|=k of the Boolean algebra B _n of all subsets of {1,...,n}, ordered by inclusion. An inequality is obtained for cutsets for A that is reminiscent of Lubell's inequality for antichains in B _n. A new result that is provided by this approach is a list of all minimum cutsets for A.Research supported in part by NSF Grant DMS 87-01475.Research supported in part by NSF Grant DMS 86-06225 and Air Force OSR-86-0076.     

A bound for certain <Emphasis Type="Italic">s</Emphasis>-extremal lattices and codes

In this paper we introduce the notion of s-extremal lattice for unimodular Type I lattices. We give a bound on the existence of certain such s-extremal lattices: an s-extremal lattice of dimension n and minimal even norm μ must satisfy n < 12μ. This result implies that such lattices are also extremal and that there are a finite number of them. We also give an equivalent bound for s-extremal self-dual codes: an s-extremal code with doubly-even minimum distance d and length n must satisfy n < 6d, moreover such codes are extremal. Received: 25 July 2006     

The existence of howell designs of siden+1 and order 2n

AHowell design of side s andorder 2n, or more briefly, anH(s, 2n), is ans×s array in which each cell either is empty or contains an unordered pair of elements from some 2n-set, sayX, such that (a) each row and each column is Latin (that is, every element ofX is in precisely one cell of each row and each column) and (b) every unordered pair of elements fromX is in at most one cell of the array. Atrivial Howell design is anH(s, 0) havingX=? and consisting of ans×s array of empty cells. A necessary condition onn ands for the existence of a nontrivialH(s, 2n) is that 0<n≦s≦2n-1. AnH(n+t, 2n) is said to contain a maximum trivial subdesign if somet×t subarray is theH(t, 0). This paper describes a recursive construction for Howell designs containing maximum trivial subdesigns and applies it to settle the existence question forH(n+1, 2n)’s: forn+1 a positive integer, there is anH(n+1, 2n) if and only ifn+1 ∉ {2, 3, 5}.     

An Anti-Ramsey Theorem

Let be the Turán number which gives the maximum size of a graph of order containing no subgraph isomorphic to . In 1973, Erdős, Simonovits and Sós [5] proved the existence of an integer such that for every integer , the minimum number of colours , such that every -colouring of the edges of which uses all the colours produces at least one all whose edges have different colours, is given by . However, no estimation of was given in [5]. In this paper we prove that for . This formula covers all the relevant values of n and p. Received January 27, 1997/Revised March 14, 2000     

On Heilbronn’s Problem in Higher Dimension

Heilbronn conjectured that given arbitrary n points in the 2-dimensional unit square [0, 1]², there must be three points which form a triangle of area at most O(1/n²). This conjecture was disproved by a nonconstructive argument of Komlós, Pintz and Szemerédi [10] who showed that for every n there is a configuration of n points in the unit square [0, 1]² where all triangles have area at least (log n/n²). Considering a generalization of this problem to dimensions d3, Barequet [3] showed for every n the existence of n points in the d-dimensional unit cube [0, 1]^d such that the minimum volume of every simplex spanned by any (d+1) of these n points is at least (1/n^d). We improve on this lower bound by a logarithmic factor (log n).     

Bounds on a generalized domination parameter

Abstract

If n is an integer, n ≥ 2 and u and v are vertices of a graph G, then u and v are said to be K_n-adjacent vertices of G if there is a subgraph of G, isomorphic to K_n , containing u and v. For n ≥ 2, a K_n- dominating set of G is a set D of vertices such that every vertex of G belongs to D or is K_n-adjacent to a vertex of D. The K_n-domination number γK_n (G) of G is the minimum cardinality among the K_n-dominating sets of vertices of G. It is shown that, for n ε {3,4}, if G is a graph of order p with no K_n-isolated vertex, then γK_n (G) ≤ p/n. We establish that this is a best possible upper bound. It is shown that the result is not true for n ≥ 5.     

Partition conditions and vertex-connectivity of graphs

It was proved ([5], [6]) that ifG is ann-vertex-connected graph then for any vertex sequencev ₁, ...,v _n ≠V(G) and for any sequence of positive integersk ₁, ...,k _n such thatk ₁+...+k _n =|V(G)|, there exists ann-partition ofV(G) such that this partition separates the verticesv ₁, ...,v(n), and the class of the partition containingv _i induces a connected subgraph consisting ofk _i vertices, fori=1, 2, ...,n. Now fix the integersk ₁, ...,k _n . In this paper we study what can we say about the vertex-connectivity ofG if there exists such a partition ofV(G) for any sequence of verticesv ₁, ...,v _n ≠V(G). We find some interesting cases when the existence of such partitions implies then-vertex-connectivity ofG, in the other cases we give sharp lower bounds for the vertex-connectivity ofG.     

Planar Ramsey numbers for cycles

For two given graphs G and H the planar Ramsey number PR(G,H) is the smallest integer n such that every planar graph F on n vertices either contains a copy of G or its complement contains a copy H. By studying the existence of subhamiltonian cycles in complements of sparse graphs, we determine all planar Ramsey numbers for pairs of cycles.     

Construction of a linear Pfaff system of Fuchsian type with algebraic manifold of singularities

We present a constructive proof of the existence of a two-dimensional completely integrable Fuchsian Pfaff system on CP ⁿ with four singular surfaces forming a pencil, with 2-step solvable monodromy group, and with fundamental solution matrix realizing a given homomorphism.     

An Explicit Construction for a Ramsey Problem

An explicit coloring of the edges of K_n is constructed such that every copy of K₄ has at least four colors on its edges. As n , the number of colors used is n^1/2+o(1). This improves upon the previous bound of O(n^2/3) due to Erds and Gyárfás obtained by probabilistic methods. The exponent 1/2 is optimal, since it is known that at least (n^1/2) colors are required in such a coloring.The coloring is related to constructions giving lower bounds for the multicolor Ramsey number r_k(C₄). It is more complicated however, because of restrictions imposed on interactions between color classes.* Research supported in part by NSF Grant No. DMS–9970325.     

On computing the number of linear extensions of a tree

An algorithm requiring O(n ²) arithmetic operations for computing the number of linear extensions of a poset whose covering graph is a tree is given.This research was partially funded by the National Science and Engineering Research Council of Canada under Grant Number A4219.     

Card shuffling and a transformation on S n

Summary We consider two card shuffling schemes. The first, which has appeared in the literature previously ([G], [RB], [T]), is as follows: start with a deck ofn cards, and pick a random tuplet { 1, 2, , n} ⁿ ; interchange cards 1 andt ₁, then interchange cards 2 andt ₂, etc. The second scheme, which can be viewed as a transformation on the symmetric groupS _n , is given by the restriction of the former shuffling scheme to tuplest which form a permutation of {1, 2,,n}.We determine the bias of each of these shuffling schemes with respect to the sets of transpositions and derangements, and the expected number of fixed points of a permutation generated by each of these shuffling schemes. For the latter scheme we prove combinatorially that the permutation which arises with the highest probability is the identity. The same question is open for the former scheme. We refute a candidate answer suggested by numerical evidence [RB].This work was carried out in part while R.S. was visiting the Institute for Mathematics and its Applications and was partly supported through NSF Grant CCR-8707539.     

Holey self-orthogonal Latin squares with symmetric orthogonal mates

We improve the existence results for holey self-orthogonal Latin squares with symmetric orthogonal mates (HSOLSSOMs) and show that the necessary conditions for the existence of a HSOLSSOM of typeh ⁿ are also sufficient with at most 28 pairs (h, n) of possible exceptions. Research supported in part by NSERC Grant A-5320 for the first author, NSF Grants CCR-9504205 and CCR-9357851 for the second author, and NSFC Grant 19231060-2 for the third author.     

Automorphisms of random graphs with specified vertices

Conditions are found under which the expected number of automorphisms of a large random labelled graph with a given degree sequence is close to 1. These conditions involve the probability that such a graph has a given subgraph. One implication is that the probability that a random unlabelledk-regular simple graph onn vertices has only the trivial group of automorphisms is asymptotic to 1 asn → ∞ with 3≦k=O(n ^1/2−c). In combination with previously known results, this produces an asymptotic formula for the number of unlabelledk-regular simple graphs onn vertices, as well as various asymptotic results on the probable connectivity and girth of such graphs. Corresponding results for graphs with more arbitrary degree sequences are obtained. The main results apply equally well to graphs in which multiple edges and loops are permitted, and also to bicoloured graphs. Research of the second author supported by U. S. National Science Foundation Grant MCS-8101555, and by the Australian Department of Science and Technology under the Queen Elizabeth II Fellowships Scheme. Current address: Mathematics Department, University of Auckland, Auckland, New Zealand.     

A regularity theorem for deformations of a compact complex surface

Let ƒ:M →D ⊑C ⁿ be a holomorphic family of compact, complex surfaces, which is locally trivial onD∖Z, for an analytic subsetZ. Conditions are found under which ƒ extends trivially toD, if the fibers of ƒ|_D∖Z are either Hirzebruch surfaces (projective bundles overP ₁), Hopf surfaces (elliptic bundles overP ₁), hyperelliptic bundles, or any compact complex surface having one of these as minimal model under blowing-down. The results of this paper are motivated by the existence of non-Hausdorff moduli spaces in the deformation of complex structure for certain complex manifolds.     

Thep-intersection number of a complete bipartite graph and orthogonal double coverings of a clique

Thep-intersection graph of a collection of finite sets {S _i } _i=1 ⁿ is the graph with vertices 1, ...,n such thati, j are adjacent if and only if |S _i S _j |p. Thep-intersection number of a graphG, herein denoted _p (G), is the minimum size of a setU such thatG is thep-intersection graph of subsets ofU. IfG is the complete bipartite graphK _n,n andp2, then _p (K _{n, n} )(n ²+(2p–1)n)/p. Whenp=2, equality holds if and only ifK _n has anorthogonal double covering, which is a collection ofn subgraphs ofK _n , each withn–1 edges and maximum degree 2, such that each pair of subgraphs shares exactly one edge. By construction,K _n has a simple explicit orthogonal double covering whenn is congruent modulo 12 to one of {1, 2, 5, 7, 10, 11}.Research supported in part by ONR Grant N00014-5K0570.