Some results on the Ryser design conjecture-III

TL;DR

This paper investigates the structure of Ryser designs, explores their properties, and provides conditions under which they are of Type-1, supporting the Ryser-Woodall conjecture that all such designs are derived from symmetric designs.

Contribution

It establishes an equivalence relation on Ryser designs, analyzes block complementation, and proves that certain conditions imply a Ryser design is of Type-1.

Findings

Ryser design with an average block size is of Type-1.

Under certain conditions, all Ryser designs are of Type-1.

Block sizes and design properties are linked to the Ryser-Woodall conjecture.

Abstract

A Ryser design $\mathcal{D}$ on $v$ points is a collection of $v$ proper subsets (called blocks) of a point-set with $v$ points such that every two blocks intersect each other in $\lambda$ points (and $\lambda < v$ is a fixed number) and there are at least two block sizes. A design $\mathcal{D}$ is called a symmetric design, if every point of $\mathcal{D}$ has the same replication number (or equivalently, all the blocks have the same size) and every two blocks intersect each other in $\lambda$ points. The only known construction of a Ryser design is via block complementation of a symmetric design. Such a Ryser design is called a Ryser design of Type-1. This is the ground for the Ryser-Woodall conjecture: "every Ryser design is of Type-1". This long standing conjecture has been shown to be valid in many situations. Let $\mathcal{D}$ denote a Ryser design of order $v$, index $\lambda$ and replication numbers $r_1,r_2$. Let $e_i$ denote the number of points of $\mathcal{D}$ with replication number $r_i$ (with $i = 1, 2$). Call a block $A$ of $\mathcal{D}$ small (respectively large) if $|A| < 2\lambda$ (respectively $|A| > 2\lambda$) and average if $|A|=2\lambda$. Let $D$ denote the integer $e_1 - r_2$ and let $\rho> 1$ denote the rational number $\dfrac{r_1-1}{r_2-1}$. Main results of the present article are the following: An equivalence relation on the set of Ryser designs is established. Some observations on the block complementation procedure of Ryser-Woodall are made. It is shown that a Ryser design with two block sizes one of which is an average block size is of Type-1. It is also shown that, under the assumption that large and small blocks do not coexist in any Ryser design equivalent to a given Ryser design, the given Ryser design must be of Type-1.