On a 3-Uniform Path-Hypergraph on 5 Vertices

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1 Applied Mathematical Sciences, Vol. 10, 2016, no. 30, HIKARI Ltd, On a 3-Uniform Path-Hypergraph on 5 Vertices Paola Bonacini Department of Mathematics and Computer Science University of Catania Viale A. Doria, 6, Catania, Italy Copyright c 2016 Paola Bonacini. This article is distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Abstract In this paper we denote by P (3) (2, 5) a path-hypergraph of rank 3 and length 3 whose vertices are {a, b, c, d, e} and edges {a, b, c}, {b, c, d} and {c, d, e}. We give iterative constructions of P (3) (2, 5)-designs of any index, determining the complete spectrum of such designs, and the spectrum of simply and strongly balanced P (3) (2, 5)-designs. In the last section we will consider the embeddings of path-designs in P (3) (2, 5)- designs. Mathematics Subject Classification: 05B05 Keywords: hypergraph, design, balanced, embedding 1 Introduction Let K v (3) = (X, E) be the complete hypergraph uniform of rank 3 defined on a set X of v vertices. In particular, E is the collection P 3 (X) of all 3-subsets of X. For any λ 1 we denote by λk v (3) the complete hypergraph uniform of rank 3 defined on X whose hyperedges are repeated λ times. Let H (3) be a subhypergraph of λk v (3). A H (3) -decomposition of λk v (3) is a pair Σ = (X, B), where B is a partition of λp 3 (X) in subsets all isomorphic to H (3). Such a decomposition is also called a H (3) -design of order v and index λ and the elements of B are called blocks of Σ. Given a H (3) -design Σ = (X, B), for any x X let us denote by d(x) the number d(x) of blocks of Σ containing x. d(x) is called degree of x.

2 1490 Paola Bonacini Definition 1.1. A H (3) -design Σ is said to be balanced if d(x) = d, for some d N, for any x X. Let A 1, A 2,..., A h be the orbits of the automorphism group of H (3) on its vertex set. Let Σ = (X, B) be a H (3) -design. Let us define the degree d Ai (x) of a vertex x X as the number of blocks of Σ containing x as an element of A i. Following [1] we have the notion of strongly balanced designs: Definition 1.2. We say that Σ = (X, B) is a strongly balanced H (3) -design if for any i {1,..., h} there exists a constant C i such that d Ai (x) = C i for any x X. Since d(x) = h i=1 d A i (x), it is obvious that a strongly balanced H (3) - design is a balanced H (3) -design. Since the converse in general does not hold, following [1, 6] we can give the following definition: Definition 1.3. We say that Σ = (X, B) is a simply balanced H (3) -design if it is a balanced H (3) -design, but not a strongly balanced H (3) -design. In this paper we will consider the following hypergraphs (see [7] for notation and further references): P (3) (2, 5), a path-hypergraph of rank 3 and length 3, whose vertices are {a, b, c, d, e} and edges {a, b, c}, {b, c, d} and {c, d, e}; we will denote it by [a, (b), c, (d), e]; P (3) (1, 5), a path-hypergraph of rank 3 and length 2, whose vertices are {a, b, c, d, e} and edges {a, b, c} and {a, c, d}; we will denote it by [b, c, (a), d, e]; P (3) (2, 4), a hypergraph (which is both a path and a star) whose vertices are {a, b, c, d} and edges {a, b, c} and {a, b, d}; we will denote it by [c, (a, b), d]. The spectrum of P (3) (2, 5)-designs of index 1 has been determined in [4] (together with the spectrum of other designs). In this paper we determine the complete spectrum of P (3) (2, 5)-designs of any index, without exceptions, and also the complete spectrum of strongly balanced and simply balanced P (3) (2, 5)-designs of index 1, again without exceptions. In the last part we study the embeddings of P (3) (2, 4) and P (3) (1, 5)-designs in a P (3) (2, 5)-design. For further results about hypergraphs designs you can see, for example, [2], [4], [5], [7] and [9] and for further results about the specific topic of balanced hypergraph designs you can see [3] and [8].

3 On a 3-uniform path-hypergraph on 5 vertices Constructions of P (3) (2, 5)-designs of any index Let us first start with the necessary conditions for the existence of a P (3) (2, 5)- designs of any index. Theorem 2.1. If Σ = (X, B) is a P (3) (2, 5)-design of order v and index λ, then: v(v 1)(v 2) B = λ and 1. if λ 0 mod 3, then v N, v 5, 2. if λ 1, 2 mod 3, then v 0, 1, 2 mod 9, v 9. Proof. Since Σ = (X, B) is a P (3) (2, 5)-design of order v and index λ, it must be: B = λ 1 ( ) v v(v 1)(v 2) = λ. 3 3 Then clearly it must be v 0, 1, 2 mod 9, if λ 1, 2 mod 3, while if λ 0 mod 3 we just need the obvious condition that v 5. In the case λ = 1 the condition v 0, 1, 2 mod 9 has been proved to be sufficient in [4]: Theorem 2.2 ([4]). For every v 0, 1, 2 mod 9, v 9, there exists a P (3) (2, 5)-design of order v and index 1. Now we are going to prove that the conditions given in Theorem 2.1 are sufficient for any λ. First we need the following construction, in such a way that, given a P (3) (2, 5)-design of order v and index λ, we will get a P (3) (2, 5)- design of order v + 1 and index λ. Note that if λ 1, 2 mod 9, we must take v 0, 1 mod 9, while if λ 0 mod 3, there are no restrictions on the order, just the obvious one that v 5. Remark 2.3. Construction v v + 1. Let Σ = (X, B) be a P (3) (2, 5)-design of order v and index λ. Let X = X { }, with / X. Given a P 4 -design Σ = (X, B ) on X of index λ (see [11]), consider for any path {x 1, x 2, x 3, x 4 } B the block [x 1, (x 2 ),, (x 3 ), x 4 ]. Then the system Σ = (X, B B ) is a P (3) (2, 5)-design of order v + 1 and index λ. Now we can prove that the conditions given in Theorem 2.1 are sufficient.

4 1492 Paola Bonacini Theorem 2.4. Let v N, v 5, and λ N. Suppose that if λ 1, 2 mod 3, then v 0, 1, 2 mod 9 and v 9. Then there exists a P (3) (2, 5)-design of order v and index λ. Proof. Let λ 1, 2 mod 3. The result follows easily from Theorem 2.2. Indeed, taken any v 0, 1, 2 mod 9, v 9, let Σ = (X, B) a P (3) (2, 5)-design of order v and index 1. Then the system Σ = (X, B ), whose blocks are those of B, each repeated λ times, is a P (3) (2, 5)-design of order v and index λ. Now let λ = 3. Let v = 5 and X = {1, 2, 3, 4, 5}. For any x X, let X \ {x} = {y 1, y 2, y 3, y 4 } and fix a path [y 1, y 2, y 3, y 4 ]. Given this fixed path we consider two blocks: [y 1, (y 2 ), x, (y 3 ), y 4 ] and [y 2, (y 4 ), x, (y 1 ), y 3 ]. Let B the set of all these blocks, obtained for any x X. It easy to see that (X, B) is a P (3) (2, 5)-design of order 5 and index 3. Now let v > 5 and λ = 3. By the construction given in Remark 2.3 the statement follows by induction for λ = 3. Let λ = 3h, for some h N. Taken any v N, v 5, by what we just proved we can consider Σ = (X, B) a P (3) (2, 5)-design of order v and index 3. Then the system Σ = (X, B ), whose blocks are those of B, each repeated h times, is a P (3) (2, 5)-design of order v and index λ. Now we are going to see that there is construction that give us a P (3) (2, 5)- design of order v 1 + v 2 and index λ, starting from two P (3) (2, 5)-designs of index λ, one of order v 1 and the other one of order v 2. Remark 2.5. Construction v 1, v 2 v 1 + v 2. Note that if λ 1, 2 mod 9, we must take v 1, v 2 0, 1 mod 9, while if λ 0 mod 3, there are no restrictions on the orders. Let Σ 1 = (X 1, B 1 ) be a P (3) (2, 5)-design of order v 1 and index λ and Σ 2 = (X 2, B 2 ) be a P (3) (2, 5)-design of order v 2 and index λ, with X 1 X 2 =. Given a P 4 -design = (X 1, B ) on X 1 of index λ (see [11]), consider for any path [x 1, x 2, x 3, x 4 ] B and y X 2 the blocks [x 1, (x 2 ), y, (x 3 ), x 4 ]. Let us call C 1 the collection of all these blocks. Similarly, given a P 4 -design = (X 2, B ) on X 2 of index λ (see [11]), consider for any path [y 1, y 2, y 3, y 4 ] B and x X 1 the blocks [y 1, (y 2 ), x, (y 3 ), y 4 ]. Let us call C 2 the collection of all these blocks. Then the system Σ = (X 1 X 2, B 1 B 2 C 1 C 2 ) is a P (3) (2, 5)-design of order v 1 + v 2 and index λ.

5 On a 3-uniform path-hypergraph on 5 vertices Balanced P (3) (2, 5)-designs In this section we will study strongly balanced and simply balanced P (3) (2, 5)- designs, determining in both cases the spectrum in the index 1 case. We will start with strongly balanced designs, as the simply balanced ones can be easily get by these constructions with some slight modifications. Let us first recall, since a P (3) (2, 5) is a hypergraph with vertices {a, b, c, d, e} and edges {a, b, c}, {b, c, d} and {c, d, e}, the orbits of the automorphisms group in a P (3) (2, 5)-design are A 1 = {a, e}, A 2 = {b, d} and A 3 = {c}. Moreover, given a P (3) (2, 5)-design Σ, for any vertex x, we denote by d Ai (x) the number of blocks of Σ containing x as an element of A i. Theorem 3.1. Let Σ = (X, B) be a strongly balanced P (3) (2, 5)-design. Then: 1. d(x) = 5(v 1)(v 2), for any x X, 2. d A1 (x) = (v 1)(v 2), d 9 A2 (x) = (v 1)(v 2) and d 9 A3 (x) = (v 1)(v 2), for any x X, 3. v 1, 2 mod 9, v 10. Proof. Let d(x) = D for any x X. Considered that the number of positions that a vertex can occupy in a block of Σ is 5, we have 5 B = D v, so that: D = 5(v 1)(v 2). In a similar way, let d 1 = d A1 (x), d 2 = d A2 (x) and d 3 = d A3 (x) for any x X. Then we have 2 B = d 1 v, 2 B = d 2 v and B = d 3 v. Again, since B = v(v 1)(v 2), we get the equalities in the statement, which imply that v 1, 2 mod 9, v 10. Remark 3.2. It is easy to see that none of the constructions given in the previous section determines a balanced P (3) (2, 5)-design. In order to prove the existence of strongly balanced P (3) (2, 5)-designs of any order v such that v 1, 2, mod 9, v 10, we will use the difference method in the case of hypergraphs of rank 3. So we need the following result (see [5] and [7] for further reference): Theorem 3.3. Let T = {x 1, x 2, x 3 } be any triple in K (3) v and let k = v If v 1, 2 mod 3, then there exist unique a, b N, with 1 a k and a b v 2a 1, such that T is obtained by translation of the triple {0, a, a + b}.

6 1494 Paola Bonacini 2. If v 0 mod 3 and T {i, k + i, 2k + i} for any i, there exist unique a, b N, with 1 a k 1 and a b v 2a 1, such that T is obtained by translation of the triple {0, a, a + b}. Proof. Let T ab = {0, a, a + b}, with either 1 a k and a b v 2a 1 if v 1, 2 mod 3 or 1 a k 1 and a b v 2a 1 if v 0 mod 3. Then it is easy to see that any other triple T a b cannot be obtained by T ab by translation. These implies that all the triples obtained by translation of the triples T ab are different. Now an easy computation shows that in this way we get all the triples in K v (3) if v 1, 2 mod 3. In the case v 0 mod 3 we get all the triples in K v (3), with the exception of those of type {i, k + i, 2k + i} for i {0,..., k 1}. This proves the statement. Now we can prove the following: Theorem 3.4. For any v 1, 2 mod 9, v 10, there exists a strongly balanced P (3) (2, 5)-design of order v and index 1. Proof. Using the previous remark, we are going to determine a strongly balanced P (3) (2, 5)-design by giving base blocks and their translated forms. Suppose that either v = 9h + 1 or v = 9h + 2, for some h 1. So any triple is the translation of a triple {0, a, a + b}, with 1 a 3h and a b v 2a 1, whose differences are a, b and v a b. In the following we will use the difference method and determine base blocks whose triples have differences {a, b, v a b} with either a = 3i + 1 and 3i + 1 b v 6i 3 or a = 3i + 2 and 3i + 2 b v 6i 5 or a = 3i + 3 and 3i + 3 b v 6i 7, with i {0,..., h 1}. Given K v (3), with set of vertices X = {0, 1,..., v 1}, consider the system Σ = (X, B), whose blocks are: 1. A ij = [3i + 3, (0), 3i + j + 1, (3i + 1), v 1], for i {0, 1,..., h 1} and j {3i + 5,..., v 6i 5} and all their translated forms, 2. B i = [0, (3i + 1), 6i + 2, (9i + 4), 12i + 6], for i {0, 1,..., h 1}, and all their translated forms, 3. C i = [v 1, (6i + 4), 3i + 1, (0), 6i + 5], for i {0, 1,..., h 1}, and all their translated forms, 4. D i = [3i + 4, (v 3i 2), 0, (3i + 1), v 3i 3], for i {0, 1,..., h 1}, and all their translated forms. Then Σ is a strongly balanced P (3) (2, 5)-design of order v and index 1.

7 On a 3-uniform path-hypergraph on 5 vertices 1495 Now we see that the spectrum of balanced design is the same of the strongly balanced ones. Theorem 3.5. Let Σ = (X, B) be a balanced P (3) (2, 5)-design. Then: 1. d(x) = 5(v 1)(v 2), for any x X, 2. v 1, 2 mod 9, v 10. Proof. The proof works as in Theorem 3.1. Now we show that there exist balanced P (3) (2, 5)-designs that are not strongly balanced. Theorem 3.6. For any v 1, 2 mod 9, v 10, there exists a simply balanced P (3) (2, 5)-design of order v and index 1. Proof. Given a block B = [a, (b), c, (d), e] in a P 3 (2, 5)-design, let us denote by B + j the translated block B + j = [a + j, (b + j), c + j, (d + j), e + j]. Consider the strongly balanced design Σ = (X, B) given in Theorem 3.4, defined on set of vertices X = {0, 1,..., v 1}. Keeping the notation of Theorem 3.4, let us consider the system Σ = (X, B ), where: B = B {B (0) 0, B (1) 0, B (2) 0, B (3) 0 } \ {B 0 + j j = 0, 1, 2, 3}, where B (0) 0 = [0, (1), 2, (3), 4], B (1) 0 = [2, (3), 5, (7), 9], B (2) 0 = [6, (3), 4, (5), 7] and B (3) 0 = [1, (2), 4, (6), 8]. So we are replacing the blocks B 0, B 0 + 1, B and B with B (0) 0, B (1) 0, B (2) 0, B (3) 0. In these last blocks we have all the triples of B 0 + j for j = 0, 1, 2, 3, just distributed in different blocks. In this way we see that Σ is a balanced design, but it is not strongly balanced. We can consider the element 3, which in the blocks B 0 + j for j = 0, 1, 2, 3 appeared once as an element of all of the three orbits, while in the blocks B (0) 0, B (1) 0, B (2) 0 and B (3) element of the orbit A 2. This means that Σ is simply balanced. 4 Embeddings in P (3) (2, 5)-designs 0 appear three times as an In this section we will study the embeddings of hypergraphs design in P (3) (2, 5)- designs. Let us denote by P (3) (2, 4) a path-hypergraph of rank 3 and length 2, whose vertices are {a, b, c, d} and edges {a, b, c} and {a, b, d}. We will denote such a hypergraph by [c, (a, b), d]. By P (3) (1, 5) we denote a path-hypergraph of rank 3 and length 2, whose vertices are {a, b, c, d, e} and edges {a, b, c} and {a, d, e}. We will denote such a hypergraph by [b, c, (a), d, e].

8 1496 Paola Bonacini Definition 4.1. A P (3) (2, 5)-design Σ = (X, B) of order v and index λ is called: P (3) (1, 5)-perfect if the family of all the P (3) (1, 5) graphs having edges {a, b, c} and {c, d, e} generates a P (3) (1, 5)-design of order v and index σ, for some σ N. In this case we say that Σ has indices (λ, µ). P (3) (2, 4)-perfect if the family of all the P (3) (2, 4) graphs having either edges {a, b, c} and {b, c, d} or {b, c, d} and {c, d, e}, not both, generates a P (3) (2, 4)-design of order v and index µ, for some µ N. In this case we say that Σ has indices (λ, σ). (P (3) (1, 5), P (3) (2, 4))-perfect if it is P (3) (1, 5)-perfect with indices (λ, µ) and P (3) (2, 4)-perfect with indices (λ, σ). In this case we say that Σ has indices (λ, µ, σ). In each of these cases we say that a system Σ (of either P (3) (1, 5) or P (3) (2, 4)) is nested in Σ. In [9], among other results, Lonc determined the complete spectrum of P (3) (1, 5)-designs (see [9, Theorem 1.3]): Theorem 4.2. There exists a P (3) (1, 5)-design of order v and index 1 if and only if v 5 and ( v 3) is even. In [2] the complete spectrum of P (3) (2, 4)-designs is determined: Theorem 4.3. There exists a P (3) (2, 4)-design of order v and index 1 if and only if v 4 and v 3 mod 4. In this section we will study the embeddings of P (3) (1, 5) and P (3) (2, 4)- designs in P (3) (2, 5)-designs. Let us first consider the necessary conditions for a P (3) (2, 5)-design to be P (3) (1, 5)-perfect, P (3) (2, 4)-perfect and (P (3) (1, 5), P (3) (2, 4))-perfect. Theorem If Σ = (X, B) is a P (3) (1, 5)-perfect P (3) (2, 5)-design of order v and indices (λ, µ), then 2λ = 3µ, B = λ v(v 1)(v 2) 9 and v If Σ = (X, B) is a P (3) (2, 4)-perfect P (3) (2, 5)-design of order v and indices (λ, σ), then 2λ = 3σ, B = λ v(v 1)(v 2) 9 and v If Σ = (X, B) is a (P (3) (1, 5), P (3) (2, 4))-perfect P (3) (2, 5)-design of order v and indices (λ, µ, σ), then 2λ = 3µ = 3σ, B = λ v(v 1)(v 2) and v 5. 9 In particular, µ = σ.

9 On a 3-uniform path-hypergraph on 5 vertices 1497 Proof. Suppose that Σ = (X, B) is a P (3) (1, 5)-perfect P (3) (2, 5)-design of order v and indices (λ, µ). Then it must be: B = λ 1 ( ) v = µ 1 ( ) v This clearly give us the statement. In the other cases the proof is similar. Let us now consider P (3) (2, 5)-designs that are P (3) (1, 5)-perfect. We prove the following: Theorem 4.5. If Σ = (X, B) is a P (3) (1, 5)-perfect P (3) (2, 5)-design of order v and indices (λ, µ), then Σ is also a (P (3) (1, 5), P (3) (2, 4))-perfect P (3) (2, 5)- design of order v and indices (λ, µ, µ). Proof. By Theorem 4.4 we can suppose that λ = 3h and µ = 2h, for some h N, h 1. If [a, (b), c, (d), e] is any block in Σ, then clearly the set of all the {b, c, d} is hk v (3). To prove the statement we need to show that for any block [a, (b), c, (d), e] Σ we can choose either {a, b, c} or {c, d, e} (not both) in such a way that together with the triple {b, c, d} we get a P (3) (2, 4)-design of order v and index 2h. For any block [a, (b), c, (d), e] let us call the triples {a, b, c} and {c, d, e} lateral triples. Choose any block B 1 B and let T 1 and T 2 its two lateral triples. Since µ = 2h, there exists another block B 2 B whose lateral triples are T 2 and T 3. If T 3 = T 1, we stop; otherwise we take another B 3 B whose lateral triples are T 3 and T 4. Continuing in this way, we determine a sequence B 1, B 2,..., B n of blocks whose lateral triples are T 1, T 2, T 2, T 3,..., T n 1, T n, T n, T 1. In these blocks, choose the triples T 1,..., T n. If n = B, then we stop. Otherwise, we choose a new block B 1 and we follow the previous procedure, determining other triples T 1,..., T n. In this way at the end of the procedure, since µ = 2h, we determine for each block B B precisely just one lateral triple and the set of these triple is hk v (3). These lateral triples, together with the central triples {b, c, d}, determine a P (3) (2, 4)-design of order v and index 2h. The converse does not hold. If Σ = (X, B) is a P (3) (2, 4)-perfect P (3) (2, 5)- design, then Σ is not, in general, P (3) (1, 5)-perfect, as we can see in the following example. Remark 4.6. Let us consider the following P (3) (2, 5)-design Σ on {0, 1, 2, 3, 4} of order 5 and index 3 having as blocks the following: [4, (2), 0, (1), 3], [0, (3), 1, (2), 4], [1, (4), 3, (2), 0], [2, (0), 3, (4), 1], [3, (2), 4, (0), 1], [3, (4), 0, (2), 1], [0, (4), 1, (2), 3], [1, (0), 2, (3), 4], [2, (1), 3, (4), 0], [2, (4), 1, (0), 3].

10 1498 Paola Bonacini If Σ were P (3) (1, 5)-perfect, then it would be of index 2 and the central triples would determine a K (3) 5. However, this is not the case, because in the central triples we see that {0, 2, 4} is repeated twice. We can see, instead, that Σ is P (3) (2, 4)-perfect of index 2. Indeed, in the blocks above the last four vertices determine a P (3) (2, 4)-design of order 5 and index 2 in Σ: [2, (0, 1), 3], [3, (1, 2), 4], [4, (3, 2), 0], [0, (3, 4), 1], [2, (4, 0), 1], [4, (0, 2), 1], [4, (1, 2), 3], [0, (2, 3), 4], [1, (3, 4), 0], [4, (1, 0), 3]. Theorem 4.7. For any v V, v 5, there exists a P (3) (1, 5)-perfect P (3) (2, 5)- design of order v and indices (3, 2). Proof. We will distinguish the cases v even and v odd. Let us first start with the case v odd and so let v = 2k + 1, k 2. We prove the statement by induction on k, the first case being k = 2, i.e. v = 5. So let v = 5. Consider the system Σ = (Z 5, B) having as blocks the following: [2, (1), 0, (3), 4], [3, (2), 1, (4), 0], [4, (3), 2, (0), 1], [0, (4), 3, (1), 2], [1, (0), 4, (2), 3], [3, (1), 0, (2), 4], [4, (2), 1, (3), 0], [0, (3), 2, (4), 1], [1, (4), 3, (0), 2], [2, (0), 4, (1), 3]. Then it is easy to see that Σ is a P (3) (1, 5)-perfect P (3) (2, 5)-design of order 5 and indices (3, 2). Let v = 2k + 1, for some k 3. By induction, let us consider Σ = (Z 2k 1, B) a P (3) (1, 5)-perfect P (3) (2, 5)-design of order v 2 and indices (3, 2). Let 1, 2 / Z 2k 1 and let us consider the following sets of blocks: B 1 = {[ r, (i), s, (i + 1), i + 2] i Z 2k 1, r s} B 2 = {[i 1, ( 1 ), i, ( 2), i + 1] i Z 2k 1 } B 3 = {[i, ( r ), i + j, (i + 2j), s ] i Z 2k 1, j = 2,..., k 1, r s}. Then, by the choice of Σ and by the difference method we see that the system Σ = (Z 2k 1 { 1, 2 }, B B 1 B 2 B 3 ) is a P (3) (1, 5)-perfect P (3) (2, 5)-design of order v and indices (3, 2). Now let v be even. We will prove this result by using what we have just proved in the case of odd order. Let v = 2k, for some k 3. Consider a system Σ = (Z 2k 1, B) which is a P (3) (1, 5)-perfect P (3) (2, 5)-design of order 2k 1 and indices (3, 2). Taken / Z 2k 1 consider the following sets of blocks: C 1 = {[i, (i + j),, (i + k), i + k + j] i Z 2k 1, j = 1,..., k 2}, C 2 = {[i, (i + k 1),, (i + k), i + 1] i Z 2k 1 }.

11 On a 3-uniform path-hypergraph on 5 vertices 1499 Then by the difference method the system Σ = (Z 2k 1 { }, B C 1 C 2 ) is a P (3) (1, 5)-perfect P (3) (2, 5)-design of order 2k and indices (3, 2). By using Remark 4.6 it is possible to prove the following result Theorem 4.8. For any v V, v 5, there exists a P (3) (2, 4)-perfect P (3) (2, 5)- design of order v and indices (3, 2) which is not P (3) (1, 5)-perfect. Proof. The proof is analogous to the one of Theorem 4.7. Indeed, we can proceed in the same way, having the only difference in the base step of the induction in the case v odd. In fact, we can choose the P (3) (2, 4)-perfect P (3) (2, 5)-design of order 5 and indices (3, 2) given in Remark 4.6, that is not a P (3) (1, 5)-perfect system. By making this choice we see that all the systems that we get in the procedure are not P (3) (1, 5)-perfect systems, but however are still P (3) (2, 4)-perfect P (3) (2, 5)-designs of order v and indices (3, 2). Now, by simply repeating the blocks obtained in the proof of Theorem 4.7 we obviously get the following result: Theorem 4.9. For any v V, v 5, and λ, µ N such that 2λ = 3µ there exists a P (3) (1, 5)-perfect P (3) (2, 5)-design of order v and indices (λ, µ). In a similar way we have: Theorem For any v V, v 5, and λ, µ N such that 2λ = 3µ there exists a P (3) (2, 4)-perfect P (3) (2, 5)-design of order v and indices (λ, µ) which is not P (3) (1, 5)-perfect. References [1] L. Berardi, M. Gionfriddo, R. Rota, Balanced and strongly balanced P k - designs, Discrete Math., 312 (2012), [2] J. C. Bermond, A. Germa, D. Sotteau, Hypergraph-designs, Ars Combinatoria, 3 (1977), [3] P. Bonacini, M. Gionfriddo, L. Marino, Construction of non-cyclic balanced P (3) (1, 5)-designs, Appl. Math. Sci., 9 (2015), [4] D. Bryant, S. Herke, B. Maenhaut, W. Wannasit, Decompositions of complete 3-uniform hypergraphs into small 3-uniform hypergraphs, Australas. J. Combin., 60 (2010), no. 2,

12 1500 Paola Bonacini [5] M. Gionfriddo, Construction of cyclic H (3) -designs, Appl. Math. Sci., 9 (2015), no. 70, [6] M. Gionfriddo, S. Küçükçifçi, L. Milazzo, Balanced and strongly balanced 4-kite designs, Util. Math., 91 (2013), [7] M. Gionfriddo, L. Milazzo, V. Voloshin, Hypergraphs and Designs, Nova Science Publishers Inc., New York, [8] M. Gionfriddo, S. Milici, Balanced P (3) (2, 4)-designs, to appear in Utilitas Mathematica, (2016). [9] Z. Lonc, Solution of a Delta-System Decomposition Problem, Journal of Combinatorial Theory A, 55 (1990), [10] M. Tarsi, Decomposition of complete multigraphs into stars, Discr. Math., 26 (1979), [11] M. Tarsi, Decomposition of a complete multigraph into simple paths: Nonbalanced handcuffed designs, J. Combinatorial Theory A, 34 (1983), no. 1, Received: January 9, 2016; Published: April 21, 2016