Colouring problems for symmetric configurations with block size 3

Erskine, Grahame; Griggs, Terry and Širáň, Jozef (2021). Colouring problems for symmetric configurations with block size 3. Journal of Combinatorial Designs, 29(6) pp. 397–423.

DOI: https://doi.org/10.1002/jcd.21773

Abstract

The study of symmetric configurations v₃ with block size 3 has a long and rich history. In this paper we consider two colouring problems which arise naturally in the study of these structures. The first of these is weak colouring, in which no block is monochromatic; the second is strong colouring, in which every block is multichromatic. The former has been studied before in relation to blocking sets. Results are proved on the possible sizes of blocking sets and we begin the investigation of strong colourings. We also show that the known 21₃ and 22₃ configurations without a blocking set are unique and make a complete enumeration of all nonisomorphic 20₃ configurations. We discuss the concept of connectivity in relation to symmetric configurations and complete the determination of the spectrum of 2‐connected symmetric configurations without a blocking set. A number of open problems are presented.

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