Combinatorial designs related to the strong perfect graph conjecture

Abstract

When α, ω are positive integers, we set n = αω + 1 and look for zero-one matrices X, Y of size n × n such that XY= YX = J − I , JX = XJ = αJ , JY = YJ = ωJ. Simple solutions of these matrix equations are easy to find; we describe ways of constructing rather messy ones. Our investigations are motivated by an intimate relationship between the pairs X, Y and minimal imperfect graphs.