Some results on the Ryser design conjecture-II

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. Such a Ryser design is called a Ryser design of Type-1. The 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 $v$ points $4\lambda -1\le v\le {\lambda }^2+\lambda +1$. 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.

Keywords:

• Ryser design
• symmetric design
• lambda

2010 Mathematics Subject Classifications:

• 05B05
• 51E05
• 62K10

Disclosure statement

No potential conflict of interest was reported by the authors.