1. Introduction
Let be the complete hypergraph, uniform of rank 3, defined on a vertex set , so that is the set of all triples of X. Let be a sub-hypergraph of . We call 3-edges the triples of V contained in the family and edges the pairs of V contained in the 3-edges of . Such pairs will be denoted by .
An H-decomposition of is a pair , where is a collection of hypergraphs all isomorphic to H that partition the edge set of . An H-decomposition is also called a H-design of order v and the elements of are called blocks.
If
is a
H-design, for any
, we call
degree of the vertex x the number
of blocks of
containing
x; for any
,
, we call
degree of the edge (see [
1]) the number
of blocks of
containing the edge
.
Given a hypergraph , there exists an induced action of the automorphism group of H on the set of the 2-subsets of the triples of . We call edge orbits the orbits of on this set.
Following the classical definition of balanced designs, it is possible to define balanced H-designs.
Definition 1. An H-design Σ is said to be balanced if the degree of each vertex is a constant.
In [
2], generalizing this idea, the concept of
edge balanced designs has been introduced:
Definition 2. An H-design is called edge balanced if for any , , the degree is constant.
We will call a balanced hypergraph design
vertex balanced, in order to make a distinction with edge balanced hypergraph designs. The concept of balanced
G-design, in the case that
G is a graph, was introduced by Hell and Rosa in [
3]. Later, a lot of work has been done in this field (see, for example, [
2,
3,
4,
5,
6,
7,
8,
9,
10,
11,
12,
13,
14]) both for graph designs and hypergraph designs.
A hypergraph is called
linear if any two 3-edges have at most one vertex in common. It is trivial to see that any
H-design, with
H a linear hypergraph, is edge balanced of constant degree
. In this paper, continuing the problem introduced in [
2], we study edge balanced
H-designs, where
H is one of the following hypergraphs:
We denote the above hypergraphs
by
,
by
,
by
,
by
and
by
. From now on by
H we will denote one of these hypergraphs. We will denote by
be the number of triples in
H, so that:
For the hypergraphs , , and we denote by A the edge orbit that corresponds to the edge ; for by A we denote the edge orbit corresponding to the edges and . Note that, for , and , the edge has degree 2 and all the others degree 1; for , the edge has degree 3 and all the others degree 1; for , the edges and have degree 2 and all of the others degree 1.
Given a hypergraph
H, a
strong vertex coloring of
H assigns distinct colors to the vertices of a 3-edge of
H. The minimum number
k such that there exists a strong vertex coloring of
H with
k colors is called
strong chromatic number of
H and denoted by
(see [
1,
15,
16]). In this paper, we consider
H-designs, with
H hypergraph with 2 or 3 edges, such that
.
In the constructions we will use the following remarks:
if X and Y are disjoint sets, with , for some , , the triples , with , , and , are all of the type , with , and , where the indices are taken ;
if X and Y are disjoint sets, with , for some , , the triples , with , , and , are all of the type either , with , , and or , with , where the indices are taken ;
if X, Y and Z are pairwise disjoint sets, such that , and , the triples , with , and , are of type , with , and .
2. Necessary Conditions
Let H be one of the hypergraphs listed before and let be an H-design. Using the previous notation, for any , :
we denote by the number of blocks containing as an element of the edge orbit A; and,
we denote by the number of blocks containing as an element of all the other edge orbits.
Proposition 1. Let be an edge balanced H-design of order v and let . Subsequently, the following conditions hold:
- (1)
for any , : - (2)
, ;
- (3)
for any ,
Proof. For any
H let
r be the number of its edges. Let:
If any edge
, for any
,
, is contained in exactly
d blocks of
, because
we have:
By the fact that
r and
m are always coprime, we immediately get that
,
, if
and
,
, if
. For any
,
, we also have:
This leads us easily to the statement. □
Remark 1. Note that, keeping the previous notation, in order to prove that an H-design of order v is edge balanced, it is sufficient to show that there exists , such that for any , .
In this paper, we want to prove that the necessary conditions for the existence of an edge balanced H-design are also sufficient:
Theorem 1. There exists an edge balanced H-design of order v if and only if either , , if or , , if .
3. Decompositions of Multipartite Hypergraphs
In this section, and for all the rest of the paper, we denote by H an hypergraph in the set . Now, we want to provide decompositions of multipartite hypergraphs that will be used in the proof of the main result. Note that this type of decomposition is possible, because the hypergraphs considered here have .
If ,...,, with , are pairwise disjoint sets, we denote by the multipartite hypergraph having as vertex set and edge set the set of all r-subsets of containing at least one vertex from every , for ,..., s.
We prove the following:
Proposition 2. Let and let X, Y and Z be three disjoint sets such that . Subsequently, there exists a decomposition of in copies of H in such a way that:
if then : if then :
Proof. Let , and . Subsequently, we get the statement by taking the following blocks:
if , then take for any ;
if , then take for any ;
if , then take for any ;
if , then take for any ;
if , then take for any ,
where the indices are takenmod . □
Now we can prove the following:
Proposition 3. Let and let , and be three disjoint sets such that . Subsequently, there exists a decomposition of in copies of H in such a way that: for any , , with .
Proof. Let , and , where for any .
Consider now
,
and
pairwise disjoint sets. When considering the following family
of paths:
where
and the indices are taken
. Note that the set:
is the edge set of
.
Let . For any path , for , by Proposition 2 we can consider a family of copies of H decomposing such that:
for any and ;
for any and .
Let . Then the blocks of provide the required decomposition of .
Let . For any path , for , by Proposition 2, we can consider a family of copies of H decomposing such that:
for any and ;
for any and .
Let . Subsequently, the blocks of provide the required decomposition of . □
4. Proof of the Main Result
Before proving Theorem 1, we need to decompose multipartite hypergraphs, as in the following result:
Proposition 4. Let and let X, Y and Z be three disjoint sets, such that and . Giventhere exists a decomposition of in copies of H in such a way that for any , , , , and : ;
.
Proof. Case 1. Let . Let , and . In this case we get the statement by taking the following family of blocks:
for
and
and for ;
for and ;
for and ;
for and .
Case 2. Let . Let , and . Let for . If , we get the statement by taking the following family of blocks for :
for
, where
(and so
)
and ;
and ;
for .
Case 3. Let
. Let
,
and
. Consider for any
:
We get the statement by taking the following family of blocks for :
for
and
and , for ;
Case 4. Let
. Let
,
and
. Consider for any
:
and let
,
,
, and
. We get the statement by taking the following family of blocks for
:
for
and
for
and
Case 5. Let
. Let
,
and
. Consider for any
:
We get the statement by taking the following family of blocks for :
for
and
and ;
and , for . □
Now we are ready to prove Theorem 1.
Proof. By Proposition 1, we just need to prove that there exists an edge balanced H-design of order v if:
and , ;
and , .
Let us first prove it in the case
, if
, and
, if
. We will use repeatedly (Theorem 3.3, [
17]).
Let
and
. Subsequently, this case follows by (Theorem 4.4, [
2]).
Let
and
. The statement follows by taking the
H-design of order 11 on
having as base blocks the following ones:
Let
and
. The statement follows by taking the
H-design of order 11 on
having as base blocks the following ones:
Let
and
. The statement follows by taking the
H-design of order 11 on
having as base blocks the following ones:
Let
and
. The statement follows by taking the
H-design of order 11 on
having as base blocks the following ones:
Now, let
, for some
,
, where:
Let us consider ,...,, Y, pairwise disjoint sets such that for and , in such a way that . We can consider the following families of blocks:
for take an edge balanced H-design of order , by what we just proved;
for any edge take a family of blocks decomposing and satisfying the conditions of Proposition 3; and,
for any , , take a family decomposing and satisfying the conditions of Proposition 4.
Let . Subsequently, it is easy to see that is an edge balanced H-design of order v. □