On some colouring of 4-cycle systems with specified block colour patterns

The present paper continues the study (begun by Quattrocchi, Colouring 4-cycle systems with specified block colour pattern: the case of embedding P3P3-designs, Electron. J. Combin. 8 (2001)) of 4-cycle systems with specified block colour patterns that are also strict colourings in the sense of Voloshin [Coloring Mixed Hypergraphs: Theory, Algorithms and Applications, American Mathematical Society, Providence RI, 2002]. Here we consider colourings with patterns of size 3 or 4 such that each colour class C   induces a P2P2-design embedded into the 4-cycle system.