Strongly Balanced 4-Kite Designs Nested into OQ-Systems

In this paper we determine the spectrum for octagon quadrangle systems [OQS] which can be partitioned into two strongly balanced 4-kitedesigns.


Introduction
Let be a graph defined on the vertex set X. Let G be a subgraph of K.A G-decomposition of K is a pair , where is a partition of the edge set of K into subsets isomorphic to K is called a G-design of order v and the classes of the partition are said to be the blocks of .A G-design is called balanced if for each vertex   x X  , the number of blocks of  containing x is a constant.Observe that if G is a regular graph then a G-design is always balanced, hence the notion of a balanced G-design becomes meaningful only for a non-regular graph G.
Let G be a graph and let 1 2 , , , h A A A  be the orbits of the automorphism group of G on its vertex-set.Let be a G-design.We define the degree as the number of blocks of  containing x as an element of A i .We say that Clearly, since for each vertex x V  the relation holds, we have that "a strongly are the edges of the 4-kite.Note that a cycle of length 3 with a pendant edge is called a 3-kite or just a kite.In the case when G is a 4-kite, a G-design is called a 4-kite-design or also a 4-kite-system.It is known that a 4-kite design of order v exists when or Further research on 4-kite designs can be found in [2].We will call the vertices a and c of the 4-kite  the lateral vertices, b the middle vertex, d the center vertex and e the terminal vertex [4][5][6].Some balanced Gdesigns, when G is a path, have been studied in [1,3].Strongly balanced G-designs were first introduced in [1], in which the spectrum of simply balanced and strongly balanced 5 and 6 -designs have been determined, where denotes a path with k vertices.

P P k
An octagon quadrangle is a graph denoted by , , , 

  v OQS
s have been defined and studied in [4,[7][8][9].In these papers, the main idea was to follow the research about hexagon triangle systems and all the others already introduced in the literature, where we can find many authors who have studied in many ways polygon triangle systems using triangulations of polygons [5,10,11].With the study of octagon quadrangles the authors have considered quadrangulations of polygons with new ideas for the research [12,13].
In what follows, if is an  , we will say that  is nesting or also that is nested in    .Similar problems, including colorings, can be found also in [14,15].
In this paper, starting from the remark that an octagon quadrangle , can be partitioned into two 4-kites , , , , , , , , the authors study OQSs which can be partitioned into two strongly balanced  

4
C e  -designs, determining their spectrum.

Necessary Existence Conditions
If is strongly balanced 4-kite design, its vertices describe four orbits in the automorphism group of a block, which is a graph 4 .We will indicate by C the number of blocks containing any vertex as a center of the 4-kite block, by T the number of blocks containing any vertex as a terminal, by L and M the number of blocks containing any vertex as lateral or median, respectively.
In this section, we determine necessary conditions for the existence of strongly balanced 4-kite designs (order v, index  ) and for the existence of OQS (order v, index  ) nesting strongly balanced 4-kite designs.These conditions are preliminary for conclusive Theorems of Section 3.
Theorem 2.1.If is a strongly balanced 4-kite design of order v and index is a strongly balanced 4-kite design of order v and index  , following the terminology described above and considering that each vertex occupies C times the central position in the blocks, necessarily: The same considerations can be done to calculate the parameters T, M, which have the same value of C. For the last parameter L, we can consider that: Thus, 1) and 2) are verified and from them 3) holds.
 be an OQS of order v and index  , nesting a strongly balanced 4-kite design

Main Existence Theorems
In what follows, if defined in Z , then the translates of B are all the blocks of type a j b j c j d j j j j j for every v j Z  .B is called a base block of  .

Theorem 3.1. There exists an OQS, of order v and index two, nesting a strongly balanced 4-kite design of index one if and only if:
1, mod10, 11.
 be an OQS of order v and index  , nesting a 4-kite design of order v and index one.From Theorem 2.2 it is  Consider the following octagon quadrangles:

B h h h h h h h B h h h h h h h B h h h h h h h B h i h h i h h i h h i B
as base blocks.This means that the blocks 1 2 belong to , , , ,  and with all their translates.
Observe that, for , the correspondent system defined in 11 1 h  Z has for blocks all the translates of the following base block: In every case, it is possible to verify that  is an OQS of order and index 10 1 we partition every block into the two 4-kites: we can verify that the collection of all the upper 4-kites forms a 4-kite-design 1 of index one.Observe also that the collection of all the lower 4-kites forms a 4-kite-design of index one.
We can verify that both the systems are strongly balanced.In fact, for them it is , and .

Conclusive Remarks and Problems
Theorem 3.1 gives completely the spectrum of OQS which can be partitioned into two strongly balanced 4kite designs.For a given v belongs to the spectrum, Theorem 3.1 gives also the method to construct an OQS of order v with the said properties.For example, if , the translates of the two base blocks: We can verify that the translates of 1,1 K and 2,1 K define a strongly balanced 4-kite designs of order 21 v  and index 1   .Further, a system of the same type and parameters is defined by the translates of 1,2 K and 2,2 K .The Theorem 3.1 permits also to find values of v for which there exist strongly balanced 4-kite designs, whose spectrum is still unknown.Thus, the statement of Theorem 3.2 can be the starting point for its determination.
In conclusion, we observe that from Theorem 3.


We can also point out that, after the determination of the spectrum, found in this paper, it is possible to study other problems about octagon quadrangle systems.It is possible to study the intersection problem among them, about which there exist an important literature, following the technique introduced in [16,17].Also, it should be interesting to examine the conjecture of Berge for linear hypergraphs, in the case in which these are OQSs, following the ideas seen in [18,19].
verify this, it is enough to consider that the system is constructed by base blocks and difference method.
1 follows the more general: Theorem 4.1.There exists an OQS, of order v and index  , nesting a strongly balanced 4-kite design of order v and of index 2