Lethbridge Number Theory and Combinatorics Seminar: Andrea Burgess

A c-colouring of a combinatorial design is an assignment of colours, chosen from a set of size c, to the points of the design. A c-colouring is equitable if every block of size k contains ⌊k/c⌋ or ⌈k/c⌉ points of each colour. In 2016, Luther and Pike characterized c-colourable balanced incomplete block designs (BIBDs), proving that nontrivial equitable c-colourings exist only for a family of highly restricted parameter sets. By contrast, equitable colourings of cycle designs seem to exist more widely. In this talk, we review known results on equitable colourings and discuss some recent results on equitable colourings of various classes of design. Motivated by the scarcity of equitably-colourable BIBDs, we introduce the concept of a semi-equitable c-colouring, where in each block of size k, one colour which does not appear at all but each other colour appears on ⌊k/(c−1)⌋ or ⌈k/(c−1)⌉ points. Reporting on joint work with William Kellough and David Pike, we give necessary conditions for the existence of semi-equitably colourable BIBDs and present construction methods for such designs using Hadamard matrices, affine planes and twin prime powers.