Asymptotic Existence of Resolvable Graph Designs

Let v ≥ k ≥ 1 and λ ≥ 0 be integers. A block design BD(v, k, λ) is a collection of k-subsets of a v-set X in which every unordered pair of elements from X is contained in exactly λ elements of . More generally, for a fixed simple graph G, a graph design GD(v, G, λ) is a collection of graphs isomorphic to G with vertices in X such that every unordered pair of elements from X is an edge of exactly λ elements of . A famous result of Wilson says that for a fixed G and λ, there exists a GD(v, G, λ) for all sufficiently large v satisfying certain necessary conditions. A block (graph) design as above is resolvable if can be partitioned into partitions of (graphs whose vertex sets partition) X. Lu has shown asymptotic existence in v of resolvable BD(v, k, λ), yet for over twenty years the analogous problem for resolvable GD(v, G, λ) has remained open. In this paper, we settle asymptotic existence of resolvable graph designs.

Set mappings defined on pairs

A set mapping on pairs over the set S is a function f such that for each unordered pair a of elements of S,f(a) is a subset of S disjoint from a. A subset H of S is said to be free for f if x∉ f({y, z}) for all x, y, z from H. In this paper, we investigate conditions imposed on the range of f which ensure that there is a large set free for f. For example, we show that if f is defined on a set of size K+ + with always |f(a)| <k then f has a free set of size K+ if the range of f satisfies the k-chain condition, or if any two sets in the range of f have an intersection of size less than θ for some θ with θ < K.

Duplication of Room squares

ARoom squareRof order 2nis a way of arranging 2nobjects (usually 1,2,…, 2n) in a square arrayRof side 2n– 1 so that:(i) every cell of the array is empty or contains two objects;(ii) each unordered pair of objects occurs once inR(iii) every row and column ofRcontains one copy of each object.

Transitive Orientation of Graphs and Identification of Permutation Graphs

The graphs considered in this paper are assumed to be finite, with no edge joining a vertex to itself and with no two distinct edges joining the same pair of vertices. An undirected graph will be denoted by G or (V, E), where V is the set of vertices and E is the set of edges. An edge joining the vertices i,j ∊ V will be denoted by the unordered pair (i,j).An orientation of G = (V, E) is an assignment of a unique direction i → j or j → i to every edge (i,j) ∊ E. The resulting directed image of G will be denoted by G→ or (V, E→), where E→ is now a set of ordered pairs E→ = {[i,j]| (i,j) ∊ E and i → j}. Notice the difference in notation (brackets versus parentheses) for ordered and unordered pairs.

An Existence Theorem for Room Squares*

It is shown that if v is an odd prime power, other than a prime of the form 22n + 1, then there exists a Room square of order v + 1.A room square of order 2n, where n is a positive integer, is an arrangement of 2n objects in a square array of 2 side 2n - 1, such that each of the (2n - 1)2 cells of the array is either-empty or contains exactly two distinct objects; each of the 2n objects appears exactly once in each row and column; and each (unordered) pair of objects occurs in exactly one cell.

Maximal Subsets of a given Set having No Triple in Common with a Steiner Triple System on the set

Let E be a finite set containing n elements, n ≡ 1, 3 (mod 6), S = S(E) a Steiner triple system on E, i.e. each unordered pair of elements of E is a subset of exactly one triple in S. Let T be a subset of E such that none of the triples of elements of T is a member of S. Erdös has asked (in a recent letter to the authors) for the maximal size of such a set T. Denote max |T| for fixed n and S by f(n, S). We prove in this note the following result:(i)(ii)for every n ≡ 1, 3 (mod 6) there exists a Steiner triple system S0 such that equality holds in i.

A Room Design of Order 14

A Room design of order 2n, where n is a positive integer, is an arrangement of 2n objects in a square array of side 2n - 1, such that each of the (2n - 1)2 cells of the array is either empty or contains exactly two distinct objects; each of the 2n objects appears exactly once in each row and column; and each (unordered) pair of objects occurs in exactly one cell. A Room design of order 2n is said to be cyclic if the entries in the (i + l) th row are obtained by moving the entries in the i th row one column to the right (with entries in the (2n - l)th column being moved to the first column), and increasing the entries in each occupied cell by l(mod 2n - 1), except that the digit 0 remains unchanged.

The number of partitions of a set of N points in k dimensions induced by hyperplanes

1. An arbitrary (k– 1)-dimensional hyperplane disconnects K-dimensional Euclidean space Ek into two disjoint half-spaces. If a set of N points in general position in Ek is given [nok +1 in a (k–1)-plane, no k in a (k–2)-plane, and so on], then the set is partitione into two subsets by the hyperplane, a point belonging to one or the other subset according to which half-space it belongs to; for this purpose the half-spaces are considered as an unordered pair.

A Room Design of Order 10

A Room design of order 2n, where n is a positive integer, is an arrangement of 2n objects in a square of side 2n - 1, so that each of the (2n - 1)2 cells of the array is either empty or contains just two distinct objects; each of the 2n objects occurs just once in each row and in each column; and each (unordered) pair of objects occurs in just one cell.