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Let (W, C) be an m-cycle system of order n and let Ω ⊂ W , |Ω| = v < n. We say that a path design (Ω, P) of order v and block size s (2 ≤ s ≤ m − 1) is embedded in (W, C) if for every p ∈ P there is an m-cycle c = (a 1 , a 2 ,. .. , a m) ∈ C such that: (1) p = [a (s − 1)-path p occurs in the m-cycle c); and (2) a k−1 , a k+s ∈ Ω. Note that in (1) and (2)(More)
Let D be the triangle with an attached edge (i. e. D is the " kite " , a graph having vertices {a 0 , a 1 , a 2 , a 3 } and edges {a 0 , a 1 }, {a 0 , a 2 }, {a 1 , a 2 }, {a 0 , a 3 }). Bermond and Schönheim [6] proved that a kite-design of order n exists if and only if n ≡ 0 or 1 (mod 8). Let (W, C) be a nontrivial kite-design of order n ≥ 8, and let V ⊂(More)
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(More)