Notes on well-distributed minimal sub-BIBDs for $\lambda=1$

TL;DR

This paper studies special balanced incomplete block designs with lambda=1 that have evenly distributed subdesigns, exploring their properties, connections to graph isomorphism, and implications for algorithmic improvements.

Contribution

It characterizes well-distributed minimal sub-BIBDs with lambda=1 and links their properties to potential enhancements in graph isomorphism algorithms.

Findings

Determined properties of subdesigns in BIBDs with lambda=1

Established connections between these BIBDs and the graph isomorphism problem

Identified design characteristics that could improve GIP algorithms

Abstract

In these notes we investigate BIBDs with $\lambda=1$ that present subdesigns evenly covering both blocks and vertices: we determine some of their basic properties, consequence of already existing results in the literature, with regards to their size and the number of intersections of pairs and triples of subdesigns of a specific kind. We also describe the link between these particular BIBDs and the graph isomorphism problem, based on Babai's paper, and point out the characteristics of these designs that would lead to improvements of the algorithm for the GIP.