Some results on the Ryser design conjecture-II

TL;DR

This paper investigates the structure of Ryser designs, provides an expression for their incidence matrix inverse, establishes necessary conditions for Type-1 classification, and supports the Ryser-Woodall conjecture with special case results.

Contribution

It introduces an explicit inverse for the incidence matrix of Ryser designs and offers partial evidence for the Ryser-Woodall conjecture, especially for designs with two block sizes.

Findings

Derived the inverse of the incidence matrix for Ryser designs.

Established necessary conditions for a Ryser design to be of Type-1.

Supported the conjecture relating parameters of Ryser designs with partial results.

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 satisfying (i) every two blocks intersect each other in $\lambda$ points for a fixed $\lambda < v$ (ii) there are at least two block sizes. A design $\mathcal{D}$ is called a symmetric design, if all the blocks of $\mathcal{D}$ have the same size (or equivalently, every point has the same replication number) 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 also known as the Ryser-Woodall complementation method. Such a Ryser design is called a Ryser design of Type-1. The Ryser-Woodall conjecture states: "every Ryser design is of Type-1". Main results of the present article are the following. An expression for the inverse of the incidence matrix $\mathsf{A}$ of a Ryser design is obtained. A necessary condition for the design to be of Type-1 is obtained. A well known conjecture states that, for a Ryser design on \textit{v} points $\mbox{ }4\lambda-1\leq v\leq\lambda^2+\lambda+1$. A partial support for this conjecture is obtained. Finally a special case of Ryser designs with two block sizes is shown to be of Type-1.