Applied Mathematical Sciences, Vol. 10, 2016, no. 30, 1489 - 1500
HIKARI Ltd, www.m-hikari.com
http://dx.doi.org/10.12988/ams.2016.512742
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
c 2016 Paola Bonacini. This article is distributed under the Creative ComCopyright mons 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
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Let Kv = (X, E) be the complete hypergraph uniform of rank 3 defined on
a set X of v vertices. In particular, E is the collection P3 (X) of all 3-subsets
(3)
of X. For any λ ≥ 1 we denote by λKv the complete hypergraph uniform of
rank 3 defined on X whose hyperedges are repeated λ times.
(3)
(3)
Let H (3) be a subhypergraph of λKv . A H (3) -decomposition of λKv is a
pair Σ = (X, B), where B is a partition of λP3 (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.
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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 A1 , A2 , . . . , Ah 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 dAi (x)
of a vertex x ∈ X as the number of blocks of Σ containing x as an element of
Ai . 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 Ci such that dAi (x) = Ci for any
x ∈ X.
Ph
(3)
Since d(x) =
i=1 dAi (x), it is obvious that a strongly balanced H
(3)
design is a balanced H -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].
On a 3-uniform path-hypergraph on 5 vertices
2
1491
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
18
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:
v(v − 1)(v − 2)
1 v
=λ
.
|B| = λ
3 3
18
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 0 =
X ∪ {∞}, with ∞ ∈
/ X. Given a P4 -design Σ0 = (X, B 0 ) on X of index λ (see
[11]), consider for any path {x1 , x2 , x3 , x4 } ∈ B0 the block [x1 , (x2 ), ∞, (x3 ), x4 ].
Then the system Σ00 = (X 0 , B ∪ B 0 ) 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.
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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 Σ0 = (X, B 0 ), 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} = {y1 , y2 , y3 , y4 } and fix a path [y1 , y2 , y3 , y4 ]. Given this fixed path
we consider two blocks:
[y1 , (y2 ), x, (y3 ), y4 ] and [y2 , (y4 ), x, (y1 ), y3 ].
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 Σ0 = (X, B 0 ), 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 v1 + v2 and index λ, starting from two P (3) (2, 5)-designs of
index λ, one of order v1 and the other one of order v2 .
Remark 2.5. Construction v1 , v2 → v1 + v2 .
Note that if λ ≡ 1, 2 mod 9, we must take v1 , v2 ≡ 0, 1 mod 9, while if
λ ≡ 0 mod 3, there are no restrictions on the orders.
Let Σ1 = (X1 , B1 ) be a P (3) (2, 5)-design of order v1 and index λ and Σ2 =
(X2 , B2 ) be a P (3) (2, 5)-design of order v2 and index λ, with X1 ∩ X2 = ∅.
Given a P4 -design ∆0 = (X1 , B 0 ) on X1 of index λ (see [11]), consider for
any path [x1 , x2 , x3 , x4 ] ∈ B 0 and y ∈ X2 the blocks [x1 , (x2 ), y, (x3 ), x4 ]. Let
us call C1 the collection of all these blocks.
Similarly, given a P4 -design ∆00 = (X2 , B 00 ) on X2 of index λ (see [11]), consider for any path [y1 , y2 , y3 , y4 ] ∈ B 00 and x ∈ X1 the blocks [y1 , (y2 ), x, (y3 ), y4 ].
Let us call C2 the collection of all these blocks.
Then the system Σ0 = (X1 ∪ X2 , B1 ∪ B2 ∪ C1 ∪ C2 ) is a P (3) (2, 5)-design of
order v1 + v2 and index λ.
1493
On a 3-uniform path-hypergraph on 5 vertices
3
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 A1 = {a, e}, A2 = {b, d} and A3 = {c}. Moreover,
given a P (3) (2, 5)-design Σ, for any vertex x, we denote by dAi (x) the number
of blocks of Σ containing x as an element of Ai .
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)
,
18
2. dA1 (x) =
x ∈ X,
for any x ∈ X,
(v−1)(v−2)
,
9
dA2 (x) =
(v−1)(v−2)
9
and dA3 (x) =
(v−1)(v−2)
,
18
for any
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)
.
18
In a similar way, let d1 = dA1 (x), d2 = dA2 (x) and d3 = dA3 (x) for any x ∈ X.
Then we have 2 · |B| = d1 · v, 2 · |B| = d2 · v and |B| = d3 · v. Again, since
, we get the equalities in the statement, which imply that
|B| = v(v−1)(v−2)
18
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):
(3)
Theorem 3.3. Let T = {x1 , x2 , x3 } be any triple in Kv
and let k = b v3 c.
1. 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}.
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Paola Bonacini
2. If v ≡ 0 mod 3 and T 6= {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 Tab = {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 Ta0 b0 cannot be obtained by Tab by
translation. These implies that all the triples obtained by translation of the
triples Tab are different.
Now an easy computation shows that in this way we get all the triples in
(3)
(3)
Kv if v ≡ 1, 2 mod 3. In the case v ≡ 0 mod 3 we get all the triples in Kv ,
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}.
(3)
Given Kv , with set of vertices X = {0, 1, . . . , v − 1}, consider the system
Σ = (X, B), whose blocks are:
1. Aij = [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. Bi = [0, (3i + 1), 6i + 2, (9i + 4), 12i + 6], for i ∈ {0, 1, . . . , h − 1}, and all
their translated forms,
3. Ci = [v − 1, (6i + 4), 3i + 1, (0), 6i + 5], for i ∈ {0, 1, . . . , h − 1}, and all
their translated forms,
4. Di = [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.
1495
On a 3-uniform path-hypergraph on 5 vertices
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)
,
18
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 Σ0 = (X, B 0 ), where:
(0)
(1)
(2)
(3)
B 0 = B ∪ {B0 , B0 , B0 , B0 } \ {B0 + j | j = 0, 1, 2, 3},
(0)
(1)
(2)
where B0 = [0, (1), 2, (3), 4], B0 = [2, (3), 5, (7), 9], B0 = [6, (3), 4, (5), 7]
(3)
and B0 = [1, (2), 4, (6), 8]. So we are replacing the blocks B0 , B0 + 1, B0 + 2
(0)
(1)
(2)
(3)
and B0 + 3 with B0 , B0 , B0 , B0 .
In these last blocks we have all the triples of B0 + j for j = 0, 1, 2, 3, just
distributed in different blocks. In this way we see that Σ0 is a balanced design,
but it is not strongly balanced. We can consider the element 3, which in the
blocks B0 + j for j = 0, 1, 2, 3 appeared once as an element of all of the three
(0)
(1)
(2)
(3)
orbits, while in the blocks B0 , B0 , B0 and B0 appear three times as an
element of the orbit A2 . This means that Σ0 is simply balanced.
4
Embeddings in P (3)(2, 5)-designs
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].
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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 Σ0 (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]):
(3)
Theorem 4.2. There
exists a P (1, 5)-design of order v and index 1 if and
v
only if v ≥ 5 and 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 6≡ 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 4.4.
1. If Σ = (X, B) is a P (3) (1, 5)-perfect P (3) (2, 5)-design of order v and inand v ≥ 5.
dices (λ, µ), then 2λ = 3µ, |B| = λ v(v−1)(v−2)
9
2. 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)
and v ≥ 5.
9
3. If Σ = (X, B) is a (P (3) (1, 5), P (3) (2, 4))-perfect P (3) (2, 5)-design of order
and v ≥ 5.
v and indices (λ, µ, σ), then 2λ = 3µ = 3σ, |B| = λ v(v−1)(v−2)
9
In particular, µ = σ.
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:
1 v
1 v
=µ
.
|B| = λ
3 3
2 3
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
(3)
the {b, c, d} is hKv . 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 B1 ∈ B and let T1 and T2 its two lateral
triples. Since µ = 2h, there exists another block B2 ∈ B whose lateral triples
are T2 and T3 . If T3 = T1 , we stop; otherwise we take another B3 ∈ B whose
lateral triples are T3 and T4 . Continuing in this way, we determine a sequence
B1 , B2 , . . . , Bn of blocks whose lateral triples are T1 , T2 , T2 , T3 ,. . . , Tn−1 , Tn ,
Tn , T1 . In these blocks, choose the triples T1 , . . . , Tn .
If n = |B|, then we stop. Otherwise, we choose a new block B10 and we
follow the previous procedure, determining other triples T10 , . . . , Tn0 0 . In this
way at the end of the procedure, since µ = 2h, we determine for each block
(3)
B ∈ B precisely just one lateral triple and the set of these triple is hKv .
These lateral triples, together with the central triples {b, c, d}, determine a
(3)
P (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].
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Paola Bonacini
If Σ were P (3) (1, 5)-perfect, then it would be of index 2 and the central triples
(3)
would determine a K5 . 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 = (Z5 , 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 =
(Z2k−1 , B) a P (3) (1, 5)-perfect P (3) (2, 5)-design of order v − 2 and indices (3, 2).
Let ∞1 , ∞2 ∈
/ Z2k−1 and let us consider the following sets of blocks:
• B1 = {[∞r , (i), ∞s , (i + 1), i + 2] | i ∈ Z2k−1 , r 6= s}
• B2 = {[i − 1, (∞1 ), i, (∞2), i + 1] | i ∈ Z2k−1 }
• B3 = {[i, (∞r ), i + j, (i + 2j), ∞s ] | i ∈ Z2k−1 , j = 2, . . . , k − 1, r 6= s}.
Then, by the choice of Σ and by the difference method we see that the system
Σ0 = (Z2k−1 ∪{∞1 , ∞2 }, B∪B1 ∪B2 ∪B3 ) 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 = (Z2k−1 , B) which is a P (3) (1, 5)-perfect P (3) (2, 5)-design of order 2k − 1
and indices (3, 2). Taken ∞ ∈
/ Z2k−1 consider the following sets of blocks:
• C1 = {[i, (i + j), ∞, (i + k), i + k + j] | i ∈ Z2k−1 , j = 1, . . . , k − 2},
• C2 = {[i, (i + k − 1), ∞, (i + k), i + 1] | i ∈ Z2k−1 }.
On a 3-uniform path-hypergraph on 5 vertices
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Then by the difference method the system Σ0 = (Z2k−1 ∪ {∞}, B ∪ C1 ∪ C2 ) 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 4.10. 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.
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Paola Bonacini
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Received: January 9, 2016; Published: April 21, 2016