Applied Mathematical Sciences, Vol. 9, 2015, no. 132, 6559 - 6566
HIKARI Ltd, www.m-hikari.com
http://dx.doi.org/10.12988/ams.2015.56441
Constructions of Octagon Quadrangle Systems1
Mario Gionfriddo and Salvatore Milici
Department of Mathematics and Computer Science
University of Catania, Italy
c 2015 Mario Gionfriddo and Salvatore Milici. This article is distributed under
Copyright the Creative Commons Attribution License, which permits unrestricted use, distribution,
and reproduction in any medium, provided the original work is properly cited.
Abstract
An octagon quadrangle is the graph having vertices x1 , x2 , ..., x8 ) and
edges {xi , xi+1 }, for every i = 1, 2, ..., 7, {x1 , x8 }, and the two chords
{x1 , x4 } and {x5 , x8 ). An octagon quadrangle system of order v, briefly
OQS(v), is a pair Σ = (X, B), where X is a finite set of v vertices and B
is a collection of edge disjoint octagon quadrangles, called blocks, which
partitions the edge set of the complete graph Kv , defined in X [2],[3]. In
this paper we give two constructions v → v + 20 for OQSs. With some
starting-systems of small order, these constructions permit to determine
the spectrum of OQS(v), with the possible exception of v = 20.
Mathematics Subject Classification: 05B05
Keywords: G-designs, Octagon Quadrangle Systems, Spectrum
1
Introduction
Let G, J be two graphs, with G subgraph of J. A G-decomposition of J is
a partition of the edge-set of J into classes each of them generates a graph
isomorph to G. If J is the complete graph Kv , a G-decomposition of Kv is
also called a G-design of order v [1],[14],[15]. In what follows we will consider
G-decompositions of Kv and G-decompositions of the bipartite complete graph
Kr,s .
An octagon quadrangle is the graph OQ having vertices x1 , x2 , ..., x8 ) and edges
{xi , xi+1 }, for every i = 1, 2, ..., 7, {x1 , x8 }, and the two chords {x1 , x4 } and
1
Reserach supported by MIUR-PRIN 2012, FIR 2015, INDAM.
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Mario Gionfriddo and Salvatore Milici
{x5 , x8 }. In what follow, we will denote such a graph by [(x1 ), x2 , x3 , (x4 ),
(x5 ), x6 , x7 , (x8 )]. We say that an OQ-design is an octagon quadrangle system
of order v, briefly an OQS(v). Therefore, equivalently, an OQS(v) is a pair
Σ = (X, B), where X is a finite set of v vertices and B is a collection of edge
disjoint octagon quadrangles (called blocks) which partitions the edge set of
the complete graph Kv defined in X.
In the recent literature many problems about G-designs have been studied.
Recent results of Catania Combinatorial unit can be found in [4],[6],[7],
[8],[9],[11]. In [2],[3],[5], the authors studied perfect OQSs, determining their
spectrum, following the direction of [13],[14],[15]. Multinestings in OQSs have
been studied in [10].
If R, S are two disjoint set of cardinality |R| = r, |S| = s, an OQ-decomposition
of Kr,s will be called also an (OQ|Kr,s )-design.
The following results and necessary existence conditions for OQSs have been
found in [7].
Theorem 1.1 : If Σ = (X, B) is an OQS of order v, then:
1) |B| = v(v−1)
; 2) v ≡ 0, 1, 5, 16 mod 20, v ≥ 16.
20
In this paper the authors determine the spectrum of OQSs, with only one
possible exception, using mainly a construction of type v 0 , v 00 −→ v 0 + v 00 . The
only exception is for v = 20. Therefore there is the following:
Open problem: Prove or disprove the existence of an OQS of order v = 20.
2
Constructions of (OQ|K5r,10s)-designs
In this section we give a construction for (OQ|K5r,10s )s.
Theorem 2.1 : If X, Y are two vertex-sets such that |X| = 5r, |Y | = 10s, for
r, s ∈ N , then it is always possible to define an (OQ|K5r,10s )-design of order
v = 5r + 10s.
Proof. Let X, Y be two vertex-sets such that |X| = 5r, |Y | = 10s, for
r, s ∈ N . Consider: 1) the partition P = {P1 , P2 , , ..., Pr } of X, into the
classes Pi = {Ai , Bi , Ci , Di , Ei }, for every i = 1, 2, ..., r; 2) the partition
Q = {Q1 , Q2 , , ..., Qs } of Y , into the classes Qj = {1j , 2j , 3j , ..., 10j }, for every
j = 1, 2, ..., s.
At this point, for every i = 1, 2, ..., r and for every j = 1, 2, ..., s, define the
collection Fi,j having the blocks:
Constructions of octagon quadrangle systems
6561
[(Ai ), 8j , Ci , (2j ), (Bi ), 5j , Di , (3j )], [(Ai ), 1j , Bi , (7j ), (Ci ), 3j , Ei , (5j )],
[(Di ), 7j , Ei , (8j ), (Bi ), 4j , Ai , (9j )], [(Ci ), 6j , Bi , (10j ), (Ei ), 2j , Di , (4j )],
[(Di ), 10j , Ai , (6j ), (Ei ), 9j , Ci , (1j )].
S
If F = j=1,2,..,h
i=1,2,..,k Fi,j , then we can verify that Ω = (X ∪ Y, F) is an OQdecomposition of the complete bipartite graph Kr,s defined in the two stable
sets X,Y .
2
3
Constructions of OQSs
The result of Section 2 permits to prove the following Theorems.
Theorem 3.1 : If Σ1 = (X1 , B1 ) is an OQS of order v 0 ≡ 0 or 5 mod 20 and
Σ2 = (X2 , B2 ) is an OQS of order v 00 ≡ 0 mod 20, then there exists an OQS
Σ of order v = v 0 + v 00 embedding both Σ0 , Σ00 .
Proof. Let Σ1 = (X1 , B1 ) and Σ2 = (X2 , B2 ) be two OQS of order respectively v 0 ∈ {20k 0 , 20k 0 +5} and v 00 = 20k 00 , for k 0 , k 00 ∈ N , such that X1 ∩X2 = ∅.
From Theorem 2.1, since v 0 = 5(4k 0 ) or v 0 = 5(4k 0 + 1), in every case v 0 is a
multiple of 5, and v 00 = 10(2k 00 ) is a multiple of 10, it is possible to define an
(OQ|Kv0 ,v00 )-design Ω = (X1 ∪ X2 , F).
At this point, if X = X1 ∪ X2 and B = B1 ∪ B2 ∪ F, examining everything, we
can verify that Σ = (X, B) is an OQS of order v = v 0 + v 00 , where it can be
v = 20(k 0 + k 00 ) or v = 20(k 0 + k 00 ) + 5.
2
Theorem 3.2 : If Σ1 = (X1 , B1 ) is an OQS of order v 0 ≡ 1 or 16 mod 20
and Σ2 = (X2 , B2 ) is an OQS of order v 00 ≡ 1 mod 20, then there exists an
OQS Σ of order v = v 0 + v 00 − 1 embedding both Σ0 , Σ00 .
Proof. Let Σ1 = (X1 , B1 ) and Σ2 = (X2 , B2 ) be two OQS of order respectively v 0 ∈ {20k 0 + 1, 20k 0 + 16} and v 00 = 20k 00 + 1, for k 0 , k 00 ∈ N , such that
|X1 ∩ X2 | =. To simplify, let X1 ∩ X2 = {∞}. From Theorem 2.1, since
v 0 = 5(4k 0 ) + 1 or v 0 = 5(4k 0 + 3) + 1, in every case v 0 − 1 is a multiple of 5, and
v 00 − 1 = 10(2k 00 ) is a multiple of 10, it is possible to define an (OQ|Kv0 −1,v00 −1 )design Ω = (X1 ∪ X2 − {∞}, F).
Therefore, as in the previous Theorem, if X = X1 ∪ X2 and B = B1 ∪ B2 ∪ F,
we can verify that Σ = (X, B) is an OQS of order v = v 0 + v 00 − 1, where it
can be v = 20(k 0 + k 00 ) + 1 or v = 20(k 0 + k 00 ) + 16.
2
4
The cases v = 16, 21, 25, 40, 60, 65
In this section we construct some particular OQS-designs, of small order, all
useful for the determination of the spectrum of OQSs, together the general
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Mario Gionfriddo and Salvatore Milici
construction.
The systems given in what follows, of order v = 16, 21, 25, 40, 60, 65, are all
contained in [7].
Theorem 4.1 : There exist OQS(v) of order v = 16.
Proof. Let v = 16. Let us consider Σ = (Z16 , B) having the blocks:
for every i ∈ {0, 1, 2, 3}, and for every j ∈ {0, 1, . . . , 7}:
Ai = [(2i), 2i + 4, 2i + 11, (2i + 5), (2i + 13), 2i + 3, 2i + 12, (2i + 8)],
Bj = [(2j + 1), 2j + 5, 2j + 3, (2j + 6), (2j + 7), 2j + 4, 2j + 10, (2j + 8)].
Then Σ is an OQS(v) of index 1. Indeed, we use again the difference method
in a way similar to the previous one and we get: 1) the differences 1, 2 and 3
in the blocks Bi ; 2) the differences 4, 5, 6 and 7 in the blocks Ai and Bi ; 3)
the difference 8 in the blocks Ai .
2
Theorem 4.2 : There exist OQS(v) of order v = 21.
Proof. Let v = 21. Let us consider Σ = (Z21 , B) whose blocks are and all
the translates of the base-block: [(18), 0, 20, (1), (16), 3, 14, (2)]. We can verify
that Σ is an OQS(v) of index 1.
2
Theorem 4.3 : There exist OQS(v) of order v = 25.
Proof. Let v = 25. Let us consider Σ = (Z24 ∪ {∞}, D), with ∞ ∈
/ Z24 ,
whose blocks are:
Ai = [(2i + 1), ∞, 2i, (2i + 3), (2i + 6), 2i + 8, 2i + 4, (2i + 5)], for every i ∈
{0, . . . , 11};
Bi = [(2i), 2i + 10k + 1, 2i + 10k + 6, (2i + 5), (2i + 10k + 7), 2i + 4, 2i + 20k +
3, (2i + 10k + 2)], for every i ∈ {0, . . . , 5};
Ci = [(2i + 8), 2i, 2i + 10, (2i + 1), (2i + 11), 2i + 3, 2i + 9, (2i + 2)], for every
i ∈ {0, . . . , 11}.
2
Theorem 4.4 : There exist OQS(v) of order v = 40.
Constructions of octagon quadrangle systems
6563
Proof. Let v = 40. Let us consider Σ = (Z13 × Z3 ∪ {∞}, B), where
∞∈
/ Z13 × Z3 and whose blocks are, for i ∈ Z13 :
[((i, 1)), (i + 1, 2), (i, 0), (∞), ((i, 2)), (i + 1, 0), (i − 1, 2), ((i + 1, 1))];
[((i + 2, 0)), (i, 0), (i + 1, 0), ((i + 5, 0)), ((i + 1, 2)), (i, 2), (i + 2, 2), ((i + 5, 2))];
[((i + 5, 1)), (i + 2, 1), (i, 1), ((i, 0)), ((i, 2)), (i + 11, 1), (i + 4, 1), ((i + 9, 1))];
[((i + 6, 0)), (i, 0), (i + 5, 0), ((i + 12, 1)), ((i + 5, 2)), (i, 2), (i + 6, 2), ((i + 10, 1))];
[((i+12, 1)), (i+6, 2), (i+9, 1), ((i, 0)), ((i+2, 1)), (i+7, 0), (i+4, 1), ((i+1, 0))];
[((i, 2)), (i + 11, 0), (i + 5, 2), ((i, 1)), ((i + 3, 2)), (i + 6, 0), (i + 12, 2), ((i + 8, 0))].
Then Σ is an OQS(v) of index 1.
2
Theorem 4.5 : There exist OQS(v) of order v = 60.
Proof. Let v = 60. Further, let Σ0 = (X, B 0 ) be an OQS(45), with X = {ai |
i = 0, . . . , 44}. Consider in Z15 :
C1 = {[(i + 5), i + 1, i, (a42 ), (i + 10), i + 4, i + 12, (i + 7)] | i = 0, . . . , 4};
C2 = {[(i + 5), i + 1, i, (a43 ), (i + 10), i + 4, i + 12, (i + 7)] | i = 5, . . . , 9};
C3 = {[(i + 5), i + 1, i, (a44 ), (i + 10), i + 4, i + 12, (i + 7)] | i = 10, . . . , 14};
C4 = {[(i + 1), a2i , i, (a2i−1 ), (i + 2), a2i−3 , i + 3, (a2i−2 )] | i = 0, . . . , 20},
where i, i + 1, i + 2, i + 3 ∈ Z15 and the indices of the aj belong to Z42 ;
C5 = {[(i + 6), a2i , i + 5, (a2i−1 ), (i + 7), a2i−3 , i + 8, (a2i−2 )] | i = 0, . . . , 20},
where i + 5, i + 6, i + 7, i + 8 ∈ Z15 and the indices of the aj belong to Z42 ;
C6 = {[(i + 11), a2i , i + 10, (a2i−1 ), (i + 12), a2i−3 , i + 13, (a2i−2 )] | i = 0, . . . , 20},
where i + 10, i + 11, i + 12, i + 13 ∈ Z15 and the indices of the aj belong to Z42 .
Then Σ = (X ∪ Z15 , B 0 ∪
S6
i=1
Ci ) is an OQS of order v = 60.
2
Theorem 4.6 : There exist OQS(v) of order v = 65.
Proof. Let v = 65. Consider Σ = (Z64 ∪ {∞}, D), with ∞ ∈
/ Z64 , whose
blocks are:
Ai = [(2i + 1), ∞, 2i, (2i + 3), (2i + 6), 2i + 8, 2i + 4, (2i + 5)], for every i ∈
{0, . . . , 31};
6564
Mario Gionfriddo and Salvatore Milici
Bi = [(2i), 2i + 10k + 1, 2i + 10k + 6, (2i + 5), (2i + 10k + 7), 2i + 4, 2i + 20k +
3, (2i + 10k + 2)], for every i ∈ {0, . . . , 15};
Cij = [(2i+5j+8), 2i, 2i+5j+10, (2i+1), (2i+5j+11), 2i+3, 2i+5j+9, (2i+2)],
for every i ∈ {0, . . . , 31} and j ∈ {0, . . . , 4}.
Then Σ is an OQS(v) of order v = 65. Indeed, by difference method, in the
blocks Ai there are the differences:
1:
2:
3:
4:
in
in
in
in
the
the
the
the
edges
edges
edges
edges
{2i + 4, 2i + 5}, {2i + 5, 2i + 6}, for every i ∈ {0, . . . , 31};
{2i + 1, 2i + 3}, {2i + 6, 2i + 8}, for every i ∈ {0, . . . , 31};
{2i, 2i + 3}, {2i + 3, 2i + 6}, for every i ∈ {0, . . . , 31};
{2i + 1, 2i + 5}, {2i + 4, 2i + 8}, for every i ∈ {0, . . . , 31}.
In the blocks Bi there are the differences:
5: given by the edges {2i, 2i + 5}, {2i + 32, 2i + 37}, {2i + 31, 2i + 36}, {2i +
33, 2i + 4}, for every i ∈ {0, . . . , 15};
31: given by the edges {2i, 2i + 31}, {2i + 32, 2i + 33}, {2i + 5, 2i + 36},
{2i + 37, 2i + 4}, for every i ∈ {0, . . . , 15};
32: given by the edges {2i, 2i + 32},{2i + 5, 2i + 37}, for every i ∈ {0, . . . , 15}.
In the blocks Cij there are the differences:
5j + 6: given by the edges {2i + 3, 2i + 5j + 9},{2i + 2, 2i + 5j + 8},
for every i ∈ {0, . . . , 31}
5j + 7: given by the edges {2i + 2, 2i + 5j + 9},{2i + 1, 2i + 5j + 8},
for every i ∈ {0, . . . , 31}
5j + 8: given by the edges {2i + 3, 2i + 5j + 11},{2i, 2i + 5j + 8},
for every i ∈ {0, . . . , 31}
5j + 9: given by the edges {2i + 1, 2i + 5j + 10},{2i + 2, 2i + 5j + 11},
for every i ∈ {0, . . . , 31}
5j + 10: given by the edges {2i, 2i + 5j + 10},{2i + 1, 2i + 5j + 11},
for every i ∈ {0, . . . , 31}, with j = 0, 1.
5
2
The spectrum of OQSs
Collecting together the results of the previous sections, it is possible to determine spectrum of the Octagon Quadruple Systems, with the possible exception
of v = 20.
Constructions of octagon quadrangle systems
6565
Theorem 5.1 : There exists an OQS of order v if and only if v ≡ 0, 1, 5, 16,
mod 20, v ≥ 16, with the possible exception of v = 20.
Proof. Let v ≡ 1 or v ≡ 16, mod 20. From Theorems 4.1, 4.2, there exist
OQS-designs of order v 0 = 16 and v 0 = 21.
Therefore, from Theorem 3.2, starting from v 0 = 16 and v 00 = 21, at first it
is possible to construct an OQS-design of order v 0 + v 00 − 1 = 36, and going
on OQS-designs of order v = 56, 76, 96, ..., all systems of order v = 20k + 16.
Similarly, starting from v 0 = 21 and v 00 = 21, at first it is possible to construct
an OQS-design of order v 0 + v 00 − 1 = 41, and going on OQS-designs of order
v = 61, 81, 101, ..., v = 20k + 1.
Let v ≡ 0 or v ≡ 5, mod 20. From Theorems 4.3, 4.4, 4.5, there exist OQSdesigns of order v 0 = 25, v 0 = 36, v 0 = 40.
Therefore, consider the Theorem 3.1. Starting from v 0 = 40 and v 00 = 40,
it is possible to construct an OQS-design of order v 0 + v 00 = 80, and going
on OQS-designs of order v = 120, 160, ..., all systems of order v = 40k + 40;
starting from v 0 = 60 and v 00 = 40, it is possible to construct OQS-designs
of order v 0 + v 00 = 100, and then OQS-designs of order v = 140, 180, ..., all
systems of order v = 60k + 40. In conclusion, we have all systems of order
v = 20k + 40. Similarly, consider always the Theorem 3.1. Starting from
v 0 = 25 and v 00 = 40, we can construct OQS-design of order v 0 + v 00 = 65,
and then of order v = 105, 145, ..., all systems of order v = 40k + 25; starting from v 0 = 45 and v 00 = 40, it is possible to construct OQS-designs of
order v 0 + v 00 = 85, and then of order v = 125, 165, ..., all systems of order
v = 40k + 25. In conclusion, we have all systems of order v = 20k + 25.
2
The case v = 20 remains a possible exception.
Open problem: Prove or disprove the existence of an OQS of order v = 20.
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Received: July 1, 2015; Published: November 4, 2015