Colouring 4 - cycle systems with specified block colour patterns : the case of embedding P 3 - designs ∗ Gaetano

A colouring of a 4-cycle system (V,B) is a surjective mapping φ : V → Γ. The elements of Γ are colours. If |Γ| = m, we have an m-colouring of (V,B). For every B ∈ B, let φ(B) = {φ(x)|x ∈ B}. There are seven distinct colouring patterns in which a 4-cycle can be coloured: type a (××××, monochromatic), type b (×××2, two-coloured of pattern 3 + 1), type c (× × 22, two-coloured of pattern 2 + 2), type d (×2 × 2, mixed two-coloured), type e (× × 24, three-coloured of pattern 2 + 1 + 1), type f (×2 ×4, mixed three-coloured), type g (×24♦, four-coloured or polychromatic). Let S be a subset of {a, b, c, d, e, f, g}. An m-colouring φ of (V,B) is said of type S if the type of every 4-cycle of B is in S. A type S colouring is said to be proper if for every type α ∈ S there is at least one 4-cycle of B having colour type α. We say that a P (v, 3, 1), (W,P), is embedded in a 4-cycle system of order n, (V,B), if every path p = [a1, a2, a3] ∈ P occurs in a 4-cycle (a1, a2, a3, x) ∈ B such that x 6∈ W . In this paper we consider the following spectrum problem: given an integer m and a set S ⊆ {b, d, f}, determine the set of integers n such that there exists a 4cycle system of order n with a proper m-colouring of type S (note that each colour class of a such colouration is the point set of a P3-design embedded in the 4-cycle system). We give a complete answer to the above problem except when S = {b}. In this case the problem is completely solved only for m = 2. AMS classification: 05B05.