Some colouring problems for Paley graphs

The Paley graph Pq, where q≡1(mod4) is a prime power, is the graph with vertices the elements of the finite field Fq and an edge between x and y if and only if x-y is a non-zero square in Fq. This paper gives new results on some colouring problems for Paley graphs and related discussion. © 2005 Elsevier B.V. All rights reserved

Squares and difference sets in finite fields

For infinitely many primes p = 4k+1 we give a slightly improved upper bound for the maximal cardinality of a set B ⊂ Z p such that the difference set B−B contains only quadratic residues. Namely, instead of the ”trivial” bound |B| ≤ √p we prove |B √p | ≤ − 1, under suitable conditions on p. The new bound is valid for approximately three quarters of the primes p = 4k + 1

Wavelength routing in optical networks of diameter two

AbstractWe consider optical networks with routing by wavelength division multiplexing. We show that wavelength switching is unnecessary in routings where communication paths use at most two edges. We then exhibit routings in some explicit pseudo-random graphs, showing that they achieve optimal performance subject to constraints on the number of edges and the maximal degree. We also observe the relative inefficiency of planar networks

Integral point sets over finite fields

We consider point sets in the affine plane $\mathbb{F}_q^2$ where each Euclidean distance of two points is an element of $\mathbb{F}_q$. These sets are called integral point sets and were originally defined in $m$-dimensional Euclidean spaces $\mathbb{E}^m$. We determine their maximal cardinality $\mathcal{I}(\mathbb{F}_q,2)$. For arbitrary commutative rings $\mathcal{R}$ instead of $\mathbb{F}_q$ or for further restrictions as no three points on a line or no four points on a circle we give partial results. Additionally we study the geometric structure of the examples with maximum cardinality.Comment: 22 pages, 4 figure

Combinatorial theorems relative to a random set

We describe recent advances in the study of random analogues of combinatorial theorems.Comment: 26 pages. Submitted to Proceedings of the ICM 201

Computational and Theoretical Aspects of \u3cem\u3eN\u3c/em\u3e-E.C. Graphs

We consider graphs with the n-existentially closed adjacency property. For a positive integer n, a graph is n-existentially closed (or n-e.c.) if for all disjoint sets of vertices A and B with \A∪ B\ = n (one of A or B can be empty), there is a vertex 2 not in A∪B joined to each vertex of A and no vertex of B. Although the n-e.c. property is straightforward to define, it is not obvious from the definition that graphs with the property exist. In 1963, Erdos and Rényi gave a non-explicit, randomized construction of such graphs. Until recently, only a few explicit families of n-e.c. graphs were known such as Paley graphs. Furthermore, n-e.c. graphs of minimum order have received much attention due to Erdos’ conjecture 011 the asymptotic order of these graphs. The exact minimum orders are only known for n = 1 and n = 2. We provide a survey of properties and examples of n-e.c. graphs. Using a computer search, a new example of a 3-e.c. graph of order 30 is presented. Previously, no known 3-e.c. graph was known to exist of that order. We give a new randomized construction of n-e.c. vertex-transitive graphs, exploiting Cayley graphs. The construction uses only elementary probability and group theory