7
1
0
6
0
6
5
4
0
7
8
9
1
3
5
2
4
6
6 5
1 0
7 2
8 7
9 8
2 9
4 3
3 4
5 6
0 1
4
6
1
3
7
8
9
5
0
2
9 8
5 9
0 6
2 1
4 3
7 5
8 7
6 0
1 2
3 4
8
7
2
1
7
8
9
0
2
4
6
1
3
5
9
8
7
3
1
2
3
4
5
6
0
7
8
9
3
2
9
8
1
9
8
7
2
3
4
5
6
0
1
8
9
7
3
4
5
6
0
1
2
9
7
8
5
4
3
9
8
7
6
0
1
2
2
3
4
5
6
0
1
7
9
8
4
5
6
0
1
2
3
8
7
9
6
0
1
2
3
4
5
9
8
7
COM BIN A TOR 1 A L
MAT HEM A TIC S
YEA R
February 1969 - June 1970
SOME THEOREMS ON COVERINGS
by
MARTIN AIGNER
Department of Statistics
University of North Carolina at Chapel Hill
Institute of Statistics Mimeo Series No. 600.2
December 1968
Research partially sponsored by the Air Force Office of Scientific Research,
Office of Aerospace Research, United States Air Force, under AFOSR Grant
No. 68-1406.
SOME THEOREMS ON COVERINGS
by
Martin Aigner
I.
INTRODUCTION
Recently several papers have appeared discussing covering and related
problems, [6, 7, 10, 11].
In this note, we treat these topics using graph-
theoretical notions.
S
Let
n
= {I, 2, ... ,
following two finite undirected graphs
T ~ (L ~ )
r,n
L ~
r,n
n}
1
and define for
T ~
r,n
~ ~ ~
with repetitions (without repetitions), and two vertices in
adjacent iff the corresponding r-tuples have at least
the
The vertex-set of
consists of all ordered (unordered) r-tuples of elements of
r,n
r
~
L ~ (T ~ )
r,n
r,n
S
n
are
coordinates (symbols)
in common.
We quote three sets of problems concerning the graphs
L ~
r,n
and
T ~
r,n
that bear special significance for coverings and designs.
(a)
Evaluation of the internal stability number
a
(b)
Evaluation of the external stability number
B.
(c)
Characterization problems.
Of these, we propose to discuss (a) and (b) in this paper.
As to (c),
attention has been focused on related association schemes and uniqueness in
terms of their parameters.
For reference, see e.g. [1, 3, 5, 8, 12], in par-
ticular [2].
Research partially sponsored by the Air Force Office of Scientific Research,
Office of Aerospace Research, United States Air Force, under AFOSR Grant
No. 68-1406.
2
THE INTERNAL STABILITY NUMBER
II.
L
Let us first investigate
£
r,n
a.
An internally stable set
M will consist
of ordered r- tu pIes in which no ordered £- tuple appears more than once.
An
orthogonal array
A
and
n
OA(r, £, A, n) with
levels is an
all possible
£ x 1
Theorem 1:
r
x
A.n
£
Proof:
- matrix where each
£
x
A.n
£
column-vectors with the same frequency
O'.(L £ ) < n £
r,n
with equality iff
Let
M be an internally stable set of
number of times an arbitrary ordered
~
constraints, strength
£, index
submatrix contains
A •
there exists an orthogonal
OA(r, £, 1, n).
array
f
r
n.
Considering all possible
L £
r,n
(£-1) - tuple
(£-1) - tuples
n
~-l
and let
appears in
f
denote the
M.
Clearly
we obtain the inequality
• n •
Equality in the above formula then means that every ordered £-tuple
appears exactly once, hence the theorem.
Examination of the frequency with which an element or coordinate can appear
leads to the following two bounds for
Theorem 2:
0'.
(L
£
r,n
).
£-1
r- 1 ,n ),
£-1
O'.(L £ ) L
r,n
i=l
O'.(L £ ) < n. O'.(L
r,n
>
M
for
£
Proof:
Given a maximal internally stable set
L
, i t is clear that
r,n
an arbitrary element can appear as, say, first coordinate at most O'.(L £-1 )
r-l,n
times.
As to the second inequality, the quantity on the right hand side which
is being subtracted plainly constitutes an upper bound for the number of r-tuples
which contain a single element at least once.
3
Let us now evaluate
~
= 1 .
a (L
1
r,n
a(L ~)
r,n
) = n,
for small values of
the set
~.
(1, ... , 1), ... (n, •.. , n)
being a
maximal internally stable set.
=
~
It is well known that the existence of an
2
equivalent to the existence of
a(L
a(L
Thus we have
n •
2
r,n
) < n
a(L 2 )
r,n
n
2
for
2
r
2
)
r,n
>
n
n+1
r-2
2
except for
n
r ~ min(pt) + 1 , where
=4
r
n
=
~ ~
= 2, 6 .
=
2
t
r
~ n+~-2.
t
=
a(L 3 ) < n 3
r,n
for
r
>
n+2
for
and
n
=
2
exists for
p
~ n+~-l,
r
Furthermore, it is known that
(p prime) and
OA(n+2, 3, 1, n)
for
•
3
~
a(L 3 )
r,n
t
n = p
odd that
n
OA(n+1, 3, 1, n)
n
3 ,n
and
more information is available, namely,
From the theory of orthogonal arrays (see [4]), we know that
and in case
is
mutually orthogonal Latin squares of order
for
For
OA(r, 2, 1, n)
=n
.
3
r
~
n+2
t
and
r
n+1
>
when
, a(L 3r,n ) = n 3 for
The corresponding theorems for the graph
T
~
r,n
n
r~
is odd,
n+1
and
read as follows, [9, 11].
Theorem 1a:
a(T ~ )
r,n
where now
n
~
r , with equality iff there exists a tactical system
Theorem 2a:
a(T ~ )
r,n
< [.!! •
r
a(T ~-1 n-1)] ,
r-1,
whence by induction
a(T ~ )
r,n
n
r
<
a(T ~ )
r,n
>
n-1
n-~+l
r-1 [ ... [ r-H1 ] ... ]]] ,
a(T ~ +1) - a(T ~-11 ) .
r, n
r- , n
TS(r,
~,
n).
4
The known values of
Q, = 1
and
a (T
n
-
2
n
) = [
r,n
r
n-l ] ]
r-l
r = 3
and
Q, = 3.
for
mod 12, r
0, 1, 3, 4
a(T
for
are:
a(T 1 ) = [ -n
r,n
r
.
Q, = 2
r = 4
Q,
for small
a
r = 2,
and
5
2
)
r,n
n
r = 3
n
-
and
n
5
mod
*
0, 1, 4, 5 mod 20,
6
and
n-l ]] _ 1
r-l
r
n - 5 mod 6
a (T
3
)
r,n
n
r
n-l
r-l
n-2 ]]]
r-2
for
r = 3,
r = 4
and
n - 1, 2, 3, 4 mod 12.
III.
THE NUMBER a
Let us define the numbers
of a set
~(L Q, )
•
and
r,n
a(T Q, )
r,n
of ordered (unordered) r-tuples
N
that each ordered (unordered) Q,-tuple
as the minimal cardinality
of elements taken from
appears at least once in
N.
S
n
such
This notion
will prove, apart from its own interest, particularly important with regard to
8, since by definition of
the evaluation of the external stability number
8 , we will be concerned with minimal sets
trary r-tuple
P
of r-tuples
coincides in at least one Q,-tuple
such that an arbi-
with some member of
P.
It is easy to derive the following theorems analogous to theorems 1 - 2a.
Theorem 3:
array
OA(r,
a (L Q, ) > n
r,n
Q,
with equality iff there exists an orthogonal
1, n).
Q"
Theorem 4:
~(L Q, ) ~ n
r,n
Q,-l
a(L Q,)
-(L Q,
) + ~ (:) a(L Q,~i
) + 1
r,n ~ a r, n-l
i=l 1
r-l, n-l
•
5
Theorem 3a:
system
a(T £ )
r,n
with equality iff there exists a tactical
TS(r, £, n).
Theorem 4a:
a(T £ )
r,n
>
)}U*)
{ ~ ~(T £-1
r
r-l, n-l
'
a(T £ )
r,n
>
{
n { n-l {.
r-l
r
.. {
whence, by induction
.. }}}
n-£+l }.
r-Hl
,
a (T £ ) < a (T £
) + ~ (T £-1
1)'
r,n =
r, n-l
r-l, nSome known values, [6, 10]:
£
=
~(L
1
2 •
1 )
r,n
r
= 3:
a(L
r
=
-
4: a(L
a(T
£
3
.
IV.
r
=
4: a(T
=n
2
)
3 ,n
2
4 ,n
2
4 ,n
)
)
3
)
4 ,n
= n2
=
{ ~ }
r
a(T 1 )
r,n
n
a(T
2
2
3 ,n
for
n
}}
{ E.3 { n-l
2
•
)
:f 2,
{ n { n-l }}
4
{
.
for
3
n { n-l
3
4"
6
n
{n-2 }}}
=
1, 2, 4, 5 mod 12 •
n - 2, 3, 4, 5 mod 6 .
2
THE EXTERNAL STABILITY NUMBER
S.
An externally stable set in an undirected graph is a set of vertices with
the property that any vertex of the graph either lies in the set or is adjacent to
at least one member of the set.
With regard to our special graphs
L £
r,n '
T £ ,we are then faced with the problem of constructing sets of r-tuples
r,n
which cover an arbitrary r-tuple
in at least an £-tuple.
of vertices adjacent to an arbitrary vertex of
(1*)
{u}
L
£ (T £ )
r,n
r,n
Since the number
clearly is
stands for the smallest integer not smaller than
u.
6
r-l
r-l
~ (~) (n-l) r-i
i=£ 1
"~
and
(r.
1 ) (nr-££)
~
. 1y~ we
respectlve
·
0b
taln
i=£
Theorem 5:
n
r
r
<
13 (L
<
r~n
(~) (n-l) r-i
~
£ )
n
(2* )
£
1
i=£
(n)
r
r
~
i=£
<
(:) (n-£)
1
r-£
13 (T £ )
r,n
<
(n)
£
( r)
£
Two more inequalities analogous to Theorems 4 and 4a are spelled out in
Theorem 6:
S(L r £,n )
~ nS(L r :-l l, n) ~
S(T £ )
<
r~n
£ = 1
of
.
S(T £ n-l) + S(T £-1
1).
r,
r-l~ n-
13 (L 1 ) = n
r,n
we could cover at most
13
Q,
= 2
,
13 (T 1 ) = [ E. ]
since for smaller values
r
r~n
r
n -1 and (n)_l vertices~ respectively.
r
~
.
2
Theorem 7:
Proof:
S(L 2 ) > --.!!r~n
= r-l .
We use induction on
r.
For
r =
2~
the theorem is trivially satisfied
since no two vertices are adjacent.
Let us take now a minimal externally stable
set
Ipl~ r~l·
2
P
in
L 2
r,n
and let us assume
appears with minimum frequency
(2*)
The inequality
13 ~ a
f
Let
x
be an element that
as a coordinate, say the first coordinate.
is true for all graphs.
7
We then have
f
Denoting by
s2' s3' .•. , sr
... ,
the second, third,
obtain
s.1 =
< f
(j =f 2)
and
... ,
n
2
=--- t
.
r-l
s
r-tuples
f
0).
s
s 2
2 => 2
Since
r-tuples
of
P
have at least
n-s
2
not containing
i-th coordinates
.
Further, any ordered
elements from above, with
s2
2nd,
of
x
... ,
r-th coordinate s
as first coordinate.
other than the
(i > 2)
S.
1
Since we
specified
above, we obtain the inequality
=
r-2
(r-I
2
n
>
(r-l - t)
+
n
+
t)2
r-2
2
-..!!- + r-l t2
r-l
r-2
thus proving the theorem.
Corollary:
arrays
Let
n
-
n
OA(r, 2, 1, [r-l ])
u mod r-l
S
provided that
u(r-l-u)
~
,
O~u <
r-l
OA(r, 2, 1, {~})
r-l
and
v , where
2
n }
r-l
{
n
(r-l)-
elements, etc., must coincide in at
least one ordered pair with some (r-l)-tuple
P
s2 -~ s J.
distinct elements appear in a combined
2
second coordinate distinct from the s3
taken from the subset of
x , we
was the minimal frequency of
f
with first coordinate distinct from the
tuple
headed by
Suppose without loss of generality
r) .
any coordinate, we note that the
f
n
<
n
the numbers of distinct elements appearing in
(t ~
r-l
total of at least
ill
r-th position of the
(i = 2,
s
<
2
and suppose orthogonal
exist, then
,
- - v mod r-l ,
o~
v <
r-l •
8
Proof:
We split the set
n-u+r-l
r-l
elements, r-l-u
S
n
into
r-l
containing
components, u
n-u
r-l
elements.
of these containing
For each one of these, we
construct an orthogonal array, thus obtaining
u(
n-u+r-l 2
n-u 2
r-l ) + (r-l-u) (r-l)
2
~
+ u(r-l-u)
r-l
r-l
< n
Now, since an arbitrary r-tuple
2
+v
r-l
n
2
r-tuples •
{ r-l }
must have at least two coordinates in the same
component, it follows that the hereby constructed set constitutes an externally
stable set, thus establishing the corollary.
Remark:
The problem of evaluating the external stability number was first
formulated in graph theory-language by Taussky-Todd
[13 ]
for the case
the general problem seems not to have been treated so far.
For
r
~
~
p
ordered p-tuples
n , let
Sp
taken from
be the minimal cardinality of a set
S
,such that every r-tuple
n
of elements in common with some member of
P
P
of un-
has at least a pair
Clearly we have
Sr
S (T 2 )
r,n
Theorem 8:
n(n-r+l)
Sp ~ p (p-l) (r-l)
Proof:
For
r
=
2 , the condition of the theorem states that every element must
appear with every other element in some p-tuple
quency of an arbitrary element is at least
tributes
p
n-l
p-l •
P , thus the minimal freSince every p-tuple
con-
appearances to the count, we obtain the inequality
S
p
>
Suppose the assertion is correct for
cardinali ty for
frequency
of
r
f , then
,
<
n(n-l)
p (p-l)
r-l
n(n-r+l)
p( p-l) (r-l) •
and let
Let
x
P
be a set of minimal
be an element of minimal
9
f
If
s
n-r+l
(p-l) (r-l)
<
denotes the number of distinct elements appearing together with
x
in
some p-tuple , we have
s
(p-l) f
<
n
r-l
s
f
Now clearly every
the
in
s
x
, t
1 - t
or
> 0
n-r+l-t (r-l)
(r-l) (p-l)
>
(r-l)-tuple
elements and
n-r+l
r-l
<
taken from the set of elements different from
must coincide in at least one pair with some p-tuple
P , and further everyone of the
s
elements appears at least
f
times.
Hence, by induction-hypothesis, we obtain
Ipi
Corollary:
implies
1
p
(n-s-l) (n-s -r+1)
(s+1) • f +
p(p-l) (r-2)
>
2
n(n-r+l)
(r-l) t
+
p(p-l) (r-l)
p(p-l) (r-2)
S(T 2 ) > n(n-r+l~
r,n
r(r-l)
n
a tactical system
Proof:
.
>
TS(r, 2, r-l)
(1)
with equality iff
n
is an integer and
r-l
exists.
From the argument used in Theorem 8, it follows that equality in (1)
t
= 0,
n
s = r-l - 1 = (r-l) . f
i. e. ,
teger and by using induction on
r
As to the sufficiency, we split
S
n
each, and construct
r-l
into
n
r-i(r-i (r-l)'
n
r-l
must be an in-
again, the necessity-part is easily established.
tactical systems
n
Hence,
r(r-l)
1)
r-l
components of
n
TS(r, 2, r-l)'
n(n-r+l)
r(r-l)
2
n
r-l
elements
thus obtaining
10
r-tuples
altogether.
Since every r-tuple
must have at least two elements in
the same component, the corollary follows.
v.
6(L
2
)
3 ,n
6(T
AND
2
3 ,n
) •
To give an application of Theorems 7 and 8, the external stability number
of
L 2
3,n
and
T 2
3,n
shall be evaluated, and properties of the minimal exter-
nally stable sets discussed.
2
{.!!-}
2
an d
P
is a minimal externally stable set
[~]2
iff it can be split into two components of
and
{¥}2
elements, respec-
tively, such that the two components have no coordinates in common and form
orthogonal arrays
OA(3, 2, 1, [
n
2"
]),
GA(3, 2, 1,
n
{2"})
upon suitable relabeling
of the elements.
Proof:
Since Latin squares exist for all
diately establishes
n
{2"}'
6.
n, the corollary to Theorem 7 immen
[ 2" ],
If we are given two Latin squares of orders
respectively, then by invoking the same corollary, we find
minimal externally stable set.
P
to be a
(The statement of the theorem simply means that,
since repetiticrs are permitted, we may relabel the coordinates arbitrarily,
making sure that the Latin squares remain coordinate-disjoint.)
an arbitrary minimal externally stable set and let
with minimum frequency
f
as a coordinate.
>
f
=
=
=
P
now be
be an element that appears
Using the same notation as in
Theorem 7, it follows that
hence
x
Let
n
2 ] •
11
Let us split
s3
P
into two components, the first containing the
coordinates which appear together with
..
. 1 es.
ma1n1ng
tr1p
Th e car d·1na1··
1t1es
0
s2
and
x , the second comprising all re-
f t h ese two sets are t h en
[ _n ] 2 , {_n}2 ,
2
2
respectively, and the same minimal frequency argument applied to one of the
s2
second coordinates (appearing with frequency
f) establishes the desired
decomposition.
To compute
and
n
odd
!3(T
2
3 ,n
),
it appears convenient to treat the cases
Let
n
be even, then
{n(n-2)}
12
for
{n(n-2)} + 1
12
Any minimal externally stable set
n
-
A, B with
A containing
n - 0, 4, 8 mod 12;
2n-I B
,
A
P
n
2'
A containing
2, 6 mod 12;
..
ta1n1ng
even
separately.
Theorem lOa:
nents
n
containing
..
conta1n1ng
n - 0, 8, 10 mod 12 •
(2)
consists of two element-disjoint compoB
n
2
n - 2, 4, 6 mod 12
containing
1,
B containing
n
2'
B
2n+ 11
e ements
n
2
containing
for
elements for
.!!+ 1
2
n
2
elements for
elements or
n - 10 mod 12,
A
such that
con-
A, B
contain all possible pairs of their respective sets of elements at least once,
and are minimal with respect to this property.
Proof:
It is a simple computation to show that the cardinality of a set
P
as described in the theorem attains the bound (2) by quoting the result of
Section III, [6].
By the argument employed in the Corollary to Theorem 8, these
sets are readily seen to be externally stable.
To give one example, suppose
sets containing
n
2
elements each.
n
= 12k + 10.
Let us split
It then follows that
S
n
into two
12
E-l
n
{l
2 •
3
{_2_} }
It remains to verify that
minimal set
P
12k
2
S
2
+ 18k + 8
can not be smaller than (2), and that every
is of the specified form.
For
n
= 2,
6 mod 12,
exact bound of Theorem 8, and so the corollary applies.
n
= 0,
4, 8 or 10 mod 12.
minimal frequency.
Let
Hence we may assume
now be such a minimal set and
P
(2) is the
f
the
S , we must have
Since we already know an upper bound for
n-2
f <--
=
4
Case A.
n - 0 rood 4
Here we have
n
f < - - 1,
n
s ~ 2f ~ 2
and thus
=4
2
(using the same nota-
tion as in Theorem 8).
Let
= E.2 -
s
(t ~
2 - t
0),
Ipi ~ ~ [
v
n - s - 1,
(s+l)f + v . {V;l}
then
J '
(3)
since the expression in parantheses gives a lower bound for the sum of the
frequencies.
Let us denote the subset of
with
x
plus the element
S
n
x Sl'
containing elements of both
consisting of the
the complementary set
Sl
and
S2
s
elements appearing
S2'
A triple of
shall be called a mixed triple.
P
(3)
now becomes
= n(n-2) + 2n[{.!} _ [.!]) + 4 (t+1) ({.!} + [.!] + 1)
12
Since for
t > 1,
12
2
2
12
2
(4) exceeds the known bound (2), we must have
>
n(n-2)
4
12
+ 12 .
(4)
2
t
0, Le.,
(5)
13
For
n
=4
mod 12,
(5) is an integer, and since we clearly cannot have
any mixed triples (we would add at least two appearances), the result follows.
For
n
= 0,
2
i. e. , two appearances to (5),
3'
we have to add
8 mod 12,
in order to obtain an integer.
Suppose we had mixed triples, then they are of
the form
(a
IS 2 1
(ab'b")
are odd and
or
(a'a"b)
Isll ~ 3
two sets
Sl
or
S2
(n ~ 8),
2
(The additional summand
Sl' b
€
E:
S2)'
but since both
we would obtain 3 additional appearances.
has cardinality
C
=5
mod 6,
and that a covering of
such sets (see [6]) contains exactly two elements appearing
Case B.
and
to (5) is accounted for by the fact that one of the
3
C-l
as all the others appear
ISll
2
C+l
2
times, where-
times. )
n _ 10 mod 12.
In this case,
With
f ~
n-2
1 - t
>
s
4
v
~
=n
1.(E.
- t) (n-2
324
2f
<
n-2
2
we obtain
- s - 1
- [.!2 D + (E. + t) • (n-2 + {.!})]
242
(6)
Comparing (6) with the previously established bound (2), we readily infer
t = 0 or 2.
Suppose t = 0,
then
=5
Is21
mod 6,
and quoting [6] again, we
see that in this case (2) cannot be improved, and by a similar argument as
before no mixed triple can occur.
For
t
= 2, the bound (2) is attained exactly,
no additional appearances are possible, hence the second possibility cited in
the theorem results.
14
Theorem lOb:
8 (T
For
2
)
3 ,n
n
odd
{n(n-l) }
12
for
n - 1, 3, 5, 7, 11 mod 12
{n(n-l)} + 1
12
n - 9
Any minimal externally stable set
components
and
A, B with element sets
P
Sl
(7)
mod 12 .
consists of two element-disjoint
and
S2' respectively, such that
A
B contain all pairs taken from their element sets at least once and are
minimal with respect to this property, where
1
Isli
or
n-l
-2-'
Sl'
n-l
2
n-3
2
=
Isli
n+l
2
Is 2 1
n;l}
S2'
1
n+3
2
Is ,
2
If one of the two sets
P
C,
Sl' S2
for
n
-
1, 5, 7
for
n
-
3, 9, 11, mod 12 .
in (8) has cardinality
C
(8)
=2
or 4 mod 6,
may contain one mixed triple with two elements from the set of cardinality
one from the other.
Proof:
Sets
P
upper bound for
as specified in (8) have cardinality (7) and thus provide an
8.
Using the notation of Theorem lOa, we then have
f
Case A.
<
n-l
4
n - 1 mod 4 .
Writing
s
n-l
= -2- -
t
IP I ~ ~[(n;l
n(n-l)
12
t ~
,
- t)
n-l
0
(n~l
f > -4- -
-
[in
equals the value of
t
[2] ,
+ en;l +
n-l
-2- +
v
t)(n~l
+
t
we obtain
,
[in J
8t-4 [~]
(9)
+ 12 2
In order not to exceed bound (7),
v
mod 12
s+l
for
t
=
t
0
= 0,
1 or 2.
Since for
t
=
1,
and vice versa, it suffices to consider
15
one of the two posibilities, e.g.,
t
=
1.
For
n
=1
mod 12 , (9) is an
integer and, since we cannot have any mixed triples (i.e., no additional appearances), the decomposition (8) results.
{n(n-l)}
12
and since
appearance at our disposal.
i f we had a triple
ances would occur.
n = 12k + 5
If
= n(n-l)
12
+
we have one additional
If there are no mixed triples, we are led to (8),
(ab'b"),
a
E
Sl'
b~b"
S2'
E
Suppose now we have a triple
with all other elements of
4
12 '
three additional appear-
(a' a"b).
Since
IA'
contains
on
S "
1
must occur
A' ,
except
a'a"
2
_ (n-l) - 16
I 24
n-3
-2- = 6k + 1
This is done as follows.
elements.
and then define
A'
b
S2' we have to construct a set of triples
which contains all possible pairs of elements, taken from
such that
Is 11 = 6k + 2,
we have
A'
S1 "
S
1
- {a"}
We construct a Steiner triple system
A"
to be:
A ll u (aI' a , a") u (a , a , a") u ... u (a 6k-l' a 6k' a ")
3
2
4
,
where
Now
A'
plainly covers all pairs distinct from
.
(n-5)
n-5
+ -412
(n-l)2 - 16
24
a'a",
and
Since we clearly cannot have more
than one mixed triple, the desired result follows.
The case
Isli
n
= 6k + 4,
= 12k + 9 can be dealt with similarly, noting that
Is 1
2
= 6k + 5 and that, since neither Sl nor S2 permits
a Steiner triple system, the bound (7) cannot be improved.
For
t
= 2,
consider the case
(9) becomes
n
n(~;l) + 1
>
{n(~;l)},
hence we only have to
= 12k + 9. Here
6k + 3 ,
6k + 6 •
No additional appearances are possible, i.e., no mixed triples can occur, and
we arrive at (8).
16
Case B.
=3
n
Here
(4)
n-3
4
f <--
=
s
n-3
=-2
n+l
v=--+t
t
2
'
and
n-3
1 [ n-l
t ) ( - - [.!.]) + (n+l + t)(n:l + [1]) J
Ipi ~3(-2-4
2
2
(4t + 2) (2[1] + 1)
n(n-l)
+
12
12
Since in this case
n(n-l)
12
is not an integer, for (10) not to exceed
the already established bound (7),
t = a or 1.
= 3,
7, 11 mod 12,
possiblities as to whether
n
(10)
We are then faced with three
and since the argument follows
the pattern of Case A, the proof has been omitted.
Let us conclude with an example to illustrate (8),
n = 15 .
18
and according to (8), there are essentially three different decompositions of
a minimal externally stable set
8
(a)
P.
no mixed triples.
2,3), (1,4,5), (1,6,7), (2,4,6), (2,5,7), (3,4,7),
(3, 5, 6»),
B
(8,9,10), (8,11,12), (8,13,14), (9,11,13), (9,12,14),
(la, 11, 14), (la, 12, 13), (8, 9, 15), (la, 11, 15), (12, 13, 15),
(13, 14, 15») •
(b)
Is11 = 7,
A
Is , = 8,
2
as in (a),
one mixed triple.
as in above minus (13,14,15),
B
mixed triple
(1, 14, 15).
(c)
Is11 = 6
Is21 = 9 ,
no mixed triple.
A = ((1,2,3), (1,2,4), (3,4,5), (3,4,6), (5,6,1),
(5, 6, 2»)
B
,
((7,8,9), (7, la, 11), (7,12,13), (7,14,15), (8, 10,12),
(8,11,14), (8,13,15), (9, la, 15), (9,11,13), (9,12,14),
(la, 13, 14), (11, 12, 15») .
17
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