| A directed candelabra t-system of order v, index λ and block size from K denoted by DCSλ(t, K, v) is a quadruple (X, S, g, B), where X is a set of v elements; S is a subset (called stem) of X with size s;g is a set of non-empty subsets (called groups) of X \S, which partition X \S; B is a family of directed subsets (called blocks) of X, and |B|∈ K for all B∈B; such that every directed t-subset T of X with |T∩(s∪Gi)|< t for all i is contained in λ blocks and no t-subset of S∩Gi for all i is contained in any block. If g has ni groups of size gi(1≤i≤r) and stem size s, then we use the notation (g1n1 g2n2 ... grnr : s) to denote group type. A directed candelabra t-system with t = 3 and K = {k} is called a directed candelabra quadruple system and denoted by DCQSλ(g1n1 g2n2...grnr : s).A directed candelabra t-system is a promotion form of a candelabra t-system. It plays an important role in the construction of directed t-systems. In this paper, we discuss the existence of a directed candelabra quadruple system by using the recursive constructions combined with direct constructions. Then we give some results as following:(1) There is a DCQSλ(gn : 0) if and only if λg(n - 1)≡0 (mod 2), λgn≡0 (mod 2), λg2n(n -1) [g (n + 1) - 3] - 0 (mod 4), n≥3, g≥1.(2) There is a DCQSλ(g2 : 0) if and only if λg≡0 (mod 2), g≥2.(3) There is a DCQSλ(g4 : s) if and only if λg≡0 (mod 2), λs≡0 (mod 2), 0≤ s<2g,g>1.(4) When λ, g, s meet the following conditions, There is a DCQSλ(g3 : s).(a)λ≡1 (mod 2), g≡0 (mod 2), s≡0 (mod 2), 0 ≤s ≤ g, g ≥ 2 ;(b)λ≡0(mod2), 0≤s≤g, g≥1.(5) When λ, g, s meet the following conditions, There is a DCQSλ(g5 : s).(a)λ≡1 (mod 2), g≡0 (mod 2), s≡0 (mod 2), 0≤s≤3g, g≥2 ;(b)λ≡0(mod2),0≤s≤3g,g≥1. |
