On balanced incomplete block designs with specified weak chromatic number

TL;DR

This paper investigates the existence of balanced incomplete block designs with a specified weak chromatic number, showing that certain necessary conditions are asymptotically sufficient for their existence under broad parameters.

Contribution

It establishes that for most parameters, the necessary conditions for weakly c-chromatic BIBDs are nearly sufficient, advancing understanding of their existence.

Findings

Necessary conditions are asymptotically sufficient for most parameters.

The case (c,k) = (2,3) is an exception.

Results apply to large values of v.

Abstract

A weak $c$-colouring of a balanced incomplete block design (BIBD) is a colouring of the points of the design with $c$ colours in such a way that no block of the design has all of its vertices receive the same colour. A BIBD is said to be weakly $c$-chromatic if $c$ is the smallest number of colours with which the design can be weakly coloured. In this paper we show that for all $c \geq 2$ and $k \geq 3$ with $(c,k) \neq (2,3)$, the obvious necessary conditions for the existence of a $(v,k,\lambda)$-BIBD are asymptotically sufficient for the existence of a weakly $c$-chromatic $(v,k,\lambda)$-BIBD.