Feasible sets for vertex colorings of P4-designs

A P4 -design of order v is a system Σ = (X, B) where X has v vertices and B is a set of copies of P4 , called blocks, decomposing the complete graph Kv on X. A BP4 -design is a P4 -design Σ endowed with a coloring of the vertices of Σ in such a way that in any block there are at least two vertices with the same color and two vertices with different colors. The feasible set Ω(Σ) of Σ is the set of integers k for which Σ is k-colorable. The minimum and maximum of Ω(Σ) are, respectively, the lower and upper chromatic number of Σ. In this paper the study of BP4 -designs is initiated, giving a lower and upper bound for feasible sets of BP4 -designs and showing the existence, for any admissible integer v, of BP4 -designs of order v with the largest possible feasible set. Some results are obtained for small values of v.