2-Regular equicolourings for P4-designs

Let GG be a graph. Then a GG-decomposition   of KvKv, a complete graph on vv vertices, is a pair Σ=(X,B)Σ=(X,B), where XX is the vertex set of KvKv and BB is a partition of the edge set of KvKv into graphs all isomorphic to GG. The elements of BB are called blocks   and ΣΣ is said to be a G-design of order  vv.In this paper we study colourings of P4-designs where, in each block of BB, two vertices are assigned the same colour and the other two another colour. We determine, among other things, families of P4-designs having a chromatic spectrum with gaps. These are the only known cases of G-designs having this property except for the families of P3-designs found by Lucia Gionfriddo.