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Embedding of finite pseudo-complements of quadrilaterals
trnal o f Statistical Planning and Inference 13 (1986) 103-110 rth-Holland
103
E M B E D D I N G OF FINITE P S E U D O - C O M P L E M E N T S OF QUADRILATERALS
M.S. MONTAKHAB Department o f Mathematics, Razi University, Bakhtaran, Iran Received 10 January 1983; revised manuscript received 13 March 1985 Recommended by D.A. Drake
Abstract: A pseudo-complement of a quadrilateral of order n is a non-trivial (n+ l)-regular linear space with n 2 - 3n + 3 points, n 2 + n - 3 lines and at least three lines of size n - 1. It is shown that if n > 23, then a pseudo-complement of a quadrilateral of order n is embeddable in a unique projective-plane of order n.
AMS Subject Classification: 05Bxx. Key words: Class; Claw; Clique; Dual; Edge; Line; Line graph; Maximal claw; Maximal clique; Point; Projective plane; Regular linear space; Simple graph; Structure; Vertex.
Introduction simple graph G consists of a non-empty finite set V(G), called the set of ver's, and a function m from the set of unordered pairs of elements of V(G) onto set {0, l } such that for every vertex P, m(P, P ) = 0. Two vertices P and Q are joinf re(P, Q) = I. Then P Q is called an edge of G. Given a vertex P of G, the number .~dges through P is called the degree of P and is denoted by d(P). Also, for two rices P and Q of G, the total number of vertices joined to both P an Q is denoted
t(P, Q). claw at a vertex P of G is an ordered pair (P, S) such that S is a subset of V(G), joined to all vertices in S, and no two vertices in S are joined. A claw (P, S) xtendable if there is a vertex Q not in S, which is joined to P and not joined to 'vertex in S. Otherwise, (P, S) is a maximal claw. set of pairwise joined vertices of G is called a clique. A clique K is a maximal ~ue if no vertex outside K is joined to all vertices in K. structure S is an ordered triplet (P, B, I ) in which P and B are non-empty disit finite sets, called the sets of points and lines, respectively, and I is a subset of B. We say a point X is contained in a line y if (X, y) belongs to L The number 3oints c o m m o n to two lines y and z is denoted by [Y, z l. Two distinct lines y and re disjoint if [y, z I= 0, otherwise they intersect. ;iven a structure S = (P, B, 1), the structure S t = (B, P, I t) is called the dual of S. :-3758/86/$3.50 © 1986, Elsevier Science Publishers B.V. (North-Holland)
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M . S . M o n t a k h a b / Finite p s e u d o - c o m p l e m e n t s o f quadrilaterals
A structure S is called a non-trivial (n + l)-regular linear space, n > 0 , if in S; (i) a point is contained in exactly n + 1 lines; (ii) two distinct points are contained in a unique c o m m o n line; (iii) no line contains all points of S. Then n is called the order of S. A projective plane of order n, n _ 2 , is a non-trivial (n + 1)-regular linear space in which all lines have the same size n + 1. A set of four lines of a projective plane of order n is called a quadrilateral if no three of them contain a common point. A pseudo-complement of a quadrilateral of order n is a non-trivial (n + 1)-regular linear space with n 2 - 3n + 3 points, n 2 + n - 3 lines and at least three lines of size n - 1. Examples of pseudo-complements of quadrilaterals of order n are obtained by removing quadrilaterals from projective planes of order n. A structure is said to be embeddable into a larger structure D if S can be extended into D by addition of new points and new lines. Mullin and Vanstone (1979) showed that if n > 2 3 , then a pseudo-complement of a quadrilateral of order n is embeddable into a projective plane of order n on the assumption that quite a complicated geometric configuration holds locally. In this paper, we improve the result above and show that if n > 23, then a pseudocomplement of a quadrilateral of order n is always embeddable into a projective plane of order n.
2. The preliminaries In Section 3 we will make use of the following results: Lemma 2.1.
If (P, S) is a claw of order s of a simple graph G, then
XeY
Proof. Let T =
V(G) \
(SU {P}). Then for Z e T,
O < e z = ½ ( m ( P ' Z ) - ~ m(X'Z))( m(P'Z)- xes ~ = m(P, Z) + ½ ~ m(X, Z)m(Z, Y ) - ~, re(P, Z)m(Z, X). YeS X¢Y
XeS
Hence
0<___zer ~ ez=d(P)-(s21) -
l(X, Y).
l(P, X) + + X~S
X, YeS X~ Y
Thus
d ( P ) > ( s 2 1 ) + ]~ I(P,X)-½ ~ i(x, Y). XeS
X, Y~S X¢ Y
[]
M.S. Montakhab / Finite pseudo-complements of quadrilaterals
105
We call a line y of a non-trivial (n ÷ 1)-regular linear space a//-line if y is of size r/--fl.
Lemma 2.2. The lines o f a pseudo-complement o f a quadrilateral S o f order n con9ist o f three l-lines, 6(n - 2) 2-lines and n 2 - 5n + 6 3-lines.
Proof. Let ai be the number of all (n - /)-lines of S. Then
ai = n2 + n - 3. i
Also, by simple counting methods,
iai = (n + 1)(n 2 - 3n + 3), i
i ( i - 1)ai = (n 2 - 3n + 3)(n 2 - 3n + 2), i
Hence ~i - n + 2 ) ( i - n + 3)ai = 6. i
~owever, S has at leastthree l-lines, and each of them contributes 2 to the left hand ;ide of the equality above. Thus a i = 0, i : / : n - 1 , n - 2 , n - 3 , and the lemma :ollows. [] It can easily be verified (see Mullin and Vanstone (1979)) that: Lemma 2.3. In a non-trivial (n + 1)-regular linear space with n 2 + n + 1 - f , f_> O, ines: (i) a fl-line is disjoint with exactly n(fl + 1 ) - f lines; (ii) a fl-line y and a ~-line z,y:#z, are mutually disjoint with exactly
n - 1)(1 - l Y, z[) + (/~ + l Y, zl)(,~ + l Y, zl) - f lines; (iii) a p o i n t P not in a fl-line y is contained in exactly fl + 1 lines disjoint with y. Using the same techniques as in the p r o o f of Theorem 3 of Mullin and Vanstone 1979), one can easily conclude that: ~ m m a 2 . 4 . / f n > 9 and y is a l-line o f a pseudo-complement o f a quadrilateral S ~f order n, then any line o f S disjoint with y is a 2-line.
It has been s h o w n by Vanstone (1973) that: ~emma 2.5. L e t S be a non-trivial (n + 1)-regular linear space. I f the number o f Joints is greater than n 2, then S is uniquely embeddable into a projective plane o f ~rder n.
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3. New points and lines In this section, we will use the well known claw-clique technique to construct new points and new lines to serve in embedding of pseudo-complements of quadrilaterals. Suppose S is a non-trivial (n+ 1)-regular linear space with n E + n + 1 - f , f->0, lines. We define a simple graph G in which V(G) is the set of lines of S and two vertices are joined if and only if the corresponding lines of S are disjoint. Then we call G the line graph of S. We call a vertex P of G a fl-vertex if its corresponding line of S is a fl-line. Then by Lemma 2.3:
Lemma 3.1. (i) I f P is a B-vertex, then d(P)=n(fl+ 1 ) - f . (ii) I f P is a fl-vertex and Q is a t~-vertex with P ¢ Q , then ~n-l-f+flt~ l(P, Q) = ~.(fl + 1)(6 + 1) - f
if m ( P , Q ) = l , otherwise.
(iii) For a fl-vertex P, there exists a claw (P, S) o f order fl + 1. Henceforth, let G be the line-graph of a pseudo-complement of a quadrilateral of order n. We will construct maximal cliques by using non-extendable claws. A maximal clique corresponds to a maximal set of mutually disjoint lines. Since every line contains at least n - 3 points, no clique can have more than n vertices unless n _<6. The lines corresponding to a clique which consists of n - 3 3-vertices and three 2-vertices cover all the points of S. The same holds for the 'lines' of a clique consisting of one l-line and n - 2 2-lines. Such cliques are, therefore, maximal. Now, by Lemmas 2.1, 2.4 and 3.1, the following lemma is immediate:
Lemma 3.2. I f n> 14, then no claw o f order 2 at a l-vertex is extendable. Lemma 3.3. I f n> 14, then a l-vertex P is contained in exactly two maximal cliques H and K o f size n - 1. Each o f these contains n - 2 2-vertices, H N K = {P} and no 2-vertex in H is joined to any 2-vertex in K. Proof. Consider a claw (P, {Q,R}). By Lemma 3.1(iii), such a claw does exist and, by Lemma 3.2, it is maximal. The set of vertices joined to P can be partitioned into three subsets M, N and T where M consists of those vertices not joined to Q, N of those not joined to R, and T o f all those joined to both Q and R. Let H = M U {P} and K = N U {P}. Clearly, R e l l , Q e K and Ht')K= {P}. Now, by Lemma 3.1(ii),
I g l - 1 + Irl--IlVl- 1+ Irl. But, by Lemma 3.1(i), Igl+lNl+lrl=2n-4. Hence, I r l = o , I g l = [ N l = n - 2 , and thus H and K are maximal cliques of size n - 1. Suppose L is another maximal clique of size n - 1 containing P. Then by Lemma
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3. l(i, ii), L and H U K have no common 2-vertex. Therefore L is of size one, a contradiction. Now, by Lemma 3.1(ii), the result follows. [] In terms of structures, Lemma 3.3 can be stated as follows:
Lemma 3.4. I f n > 14, then a l-line y o f a pseudo-complement o f a quadrilateral o f order n is contained in exactly two classes H and K o f size n - 1 such that: (i) each class contains n - 2 2-lines; (ii) H N K = { y } ; (iii) two lines are in the same class i f and only i f they are disjoint.
By Lemmas 2.3(iii) and 3.4 it follows: Corollary 3.5. Let y and z be two distinct l-lines o f a pseudo-complement o f a quadrilateral o f order n. I f n > 14, then every 2-line disjoint with y is disjoint with exactly one 2-line in every class containing z.
Corollary 3.6. Let y be a l-line o f a pseudo-complement o f a quadrilateral o f order n. I f n > 14, then every 3-line is disjoint with exactly two lines in every class containing y.
Let S be a pseudo-complement of a quadrilateral of order n. We consider its linegraph G again. Then, by Lemmas 2.1, 3.1, 3.4 and Corollary 3.5:
Lemma 3.7. Let P be a 2-vertex o f G and n> 19. Then there exists a claw (P, {R1,R2,R3} ) in which R 1 is a 1-vertex and the others are 2-vertices. Furthermore, such a claw is not extendable.
Lemma 3.8. Let P be a 2-vertex o f G and n>21. Then G cannot have a claw (P, {R1,R2,R3,R4} ) in which R l is a 1-vertex, R 2 is a 2-vertex, and each o f the others is a 3-vertex.
Proof. Suppose such a claw does exist and let Hi, 1 _
But, by Lemma 3.1 and Corollaries 3.5, 3.6,
[H~NH21=O,
IHINH3[=IH, NH4I= I,
IH2NH~I-<7,
IH2NH4I <- 7,
IH3NHaI-< l l .
Hence n+l
i l
-`22
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M.S. Montakhab / Finite pseudo-complements of quadrilaterals
so that n-<21, a contradiction.
[]
Lemma 3.9. I f n > 23, then every 2-vertex P is contained in exactly three maximal cliques Ki, i= 1, 2, 3, o f size n - 1, where K l consists o f a unique 1-vertex and n - 2 2-vertices, and each o f the others consists o f three 2-vertices and n - 3 3-vertices. Furthermore, K i f3 Kj = {P }, 1 -
A = ~ m ( P , X ) ( 1 - m ( R I , X ) - m ( R E , X)). XeT
It is easily seen that the contribution of each vertex X to A is 1 if X e N a n d is nonpositive otherwise. Hence, by Lemma 3.1(i),(ii),
INI>-A=n-2. Thus, if/(3 is a maximal clique including {P} ON, then lgal ___n- 1. In a similar fashion, one can prove that R2 is contained in a maximal clique/(2 of size at least n-1. Suppose X E ( K i N K j ) \ {P}, l
Igi u gj I-< I(P, X ) + 2 < n + 3,
Igingil -
we have
2(n- 1)-< IK l + IK l- 2 3 and the structure corresponding to G has n 2 - 3 n + 3 points, we have ]Ki[ 23. Then P is contained in exactly four
M.S. Montakhab / Finite pseudo-complements of quadrilaterals
109
maximal cliques K i, 1 <_i <_4, o f size n. Each consists o f three 2-vertices and n - 3 3-vertices. Furthermore,
inKj--{P}, Proof. Let Q be a 1-vertex and H~ and H2 be the maximal cliques of size n - 1 containing Q. Then, by Corollary 3.6, P is joined to exactly two 2-vertices in both HI and HE. Thus, by Lemma 3.9, P is contained in at least four maximal cliques Ki, 1 < i < 4, of size n each of which consists of three 2-vertices and n - 3 3-vertices. Now, by using the same techniques as in Lemma 3.9, the result follows. [] In terms of structures, Lemmas 3.9 and 3.10 can be stated as follows:
Lemma 3.11. A 2-line y o f a pseudo-complement o f a quadrilateral o f order n, n>23, is contained in exactly three classes Ki, 1 23, is contained in exactly f o u r classes Ki, 1 < i <_4, o f size n such that: (i) each Ki consists o f three 2-lines and n - 3 3-lines; (ii) K i N K y = { y }, l_
2oroHary 3.13. Let S be a pseudo-complement o f a quadrilateral o f order n, n > 23. then every point o f S is contained in a unique line o f every class described in ~emmas 3.4, 3.11 and 3.12. ~,emma 3.14. A pseudo-complement S o f a quadrilateral o f order n, n > 23, has :xactly 4 n - 2 classes o f the type described in Lemmas 3.4, 3.11 and 3.12. 'roof. We count in two ways, the number of all ordered pairs (y, K) where y is an -line of S, i = 2, 3, and K is a class of size n which contains y. If a is the number ff classes of size n, then by Lemmas 3.11 and 3.12, 1 2 ( n - 2 ) + 4 ( n 2 - 5 n + 6 ) = n a , vhence ct = 4 n - 8 . The result follows by Lemma 3.4. []
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4. Embedding Let S be a pseudo-complement of a quadrilateral of order n, n > 23. If we add as new points all classes described in Lemmas 3.4, 3.11 and 3.12, to every line contained in them, then by Lemmas 3.13 and 3.14, we get a larger structure D whose dual D t is a non-trivial (n+ D-regular linear space with n 2 + n - 3 points and n2+ n + 1 lines. By 2.5, D t is uniquely embeddable into a projective plane of order n and the dual of a finite projective plane is also a finite projective plane. Thus:
Theorem. A pseudo-complement o f a quadrilateral of order n, n > 23, is uniquely embeddable into a projective plane o f order n.
Acknowledgement I would like to thank the referee for pointing out several errors and for other valuable suggestions.
References Hughes, D.R. and F.C. Piper (1973). Projective Planes, Springer, Berlin-Heidelberg-New York. Mullin R.C. and S.A. Vanstone (1979). Embedding the pseudo-complement of a quadrilateral in a finite projective plane, Ann. New York Acad. Sci. 319, 405--412. Vanstone, S.A. (1973). The extendibility of (r, 1)-designs, Proc. Third Manitoba Conference on Numerical Math., 409-418.
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E M B E D D I N G OF FINITE P S E U D O - C O M P L E M E N T S OF QUADRILATERALS
M.S. MONTAKHAB Department o f Mathematics, Razi University, Bakhtaran, Iran Received 10 January 1983; revised manuscript received 13 March 1985 Recommended by D.A. Drake
Abstract: A pseudo-complement of a quadrilateral of order n is a non-trivial (n+ l)-regular linear space with n 2 - 3n + 3 points, n 2 + n - 3 lines and at least three lines of size n - 1. It is shown that if n > 23, then a pseudo-complement of a quadrilateral of order n is embeddable in a unique projective-plane of order n.
AMS Subject Classification: 05Bxx. Key words: Class; Claw; Clique; Dual; Edge; Line; Line graph; Maximal claw; Maximal clique; Point; Projective plane; Regular linear space; Simple graph; Structure; Vertex.
Introduction simple graph G consists of a non-empty finite set V(G), called the set of ver's, and a function m from the set of unordered pairs of elements of V(G) onto set {0, l } such that for every vertex P, m(P, P ) = 0. Two vertices P and Q are joinf re(P, Q) = I. Then P Q is called an edge of G. Given a vertex P of G, the number .~dges through P is called the degree of P and is denoted by d(P). Also, for two rices P and Q of G, the total number of vertices joined to both P an Q is denoted
t(P, Q). claw at a vertex P of G is an ordered pair (P, S) such that S is a subset of V(G), joined to all vertices in S, and no two vertices in S are joined. A claw (P, S) xtendable if there is a vertex Q not in S, which is joined to P and not joined to 'vertex in S. Otherwise, (P, S) is a maximal claw. set of pairwise joined vertices of G is called a clique. A clique K is a maximal ~ue if no vertex outside K is joined to all vertices in K. structure S is an ordered triplet (P, B, I ) in which P and B are non-empty disit finite sets, called the sets of points and lines, respectively, and I is a subset of B. We say a point X is contained in a line y if (X, y) belongs to L The number 3oints c o m m o n to two lines y and z is denoted by [Y, z l. Two distinct lines y and re disjoint if [y, z I= 0, otherwise they intersect. ;iven a structure S = (P, B, 1), the structure S t = (B, P, I t) is called the dual of S. :-3758/86/$3.50 © 1986, Elsevier Science Publishers B.V. (North-Holland)
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M . S . M o n t a k h a b / Finite p s e u d o - c o m p l e m e n t s o f quadrilaterals
A structure S is called a non-trivial (n + l)-regular linear space, n > 0 , if in S; (i) a point is contained in exactly n + 1 lines; (ii) two distinct points are contained in a unique c o m m o n line; (iii) no line contains all points of S. Then n is called the order of S. A projective plane of order n, n _ 2 , is a non-trivial (n + 1)-regular linear space in which all lines have the same size n + 1. A set of four lines of a projective plane of order n is called a quadrilateral if no three of them contain a common point. A pseudo-complement of a quadrilateral of order n is a non-trivial (n + 1)-regular linear space with n 2 - 3n + 3 points, n 2 + n - 3 lines and at least three lines of size n - 1. Examples of pseudo-complements of quadrilaterals of order n are obtained by removing quadrilaterals from projective planes of order n. A structure is said to be embeddable into a larger structure D if S can be extended into D by addition of new points and new lines. Mullin and Vanstone (1979) showed that if n > 2 3 , then a pseudo-complement of a quadrilateral of order n is embeddable into a projective plane of order n on the assumption that quite a complicated geometric configuration holds locally. In this paper, we improve the result above and show that if n > 23, then a pseudocomplement of a quadrilateral of order n is always embeddable into a projective plane of order n.
2. The preliminaries In Section 3 we will make use of the following results: Lemma 2.1.
If (P, S) is a claw of order s of a simple graph G, then
XeY
Proof. Let T =
V(G) \
(SU {P}). Then for Z e T,
O < e z = ½ ( m ( P ' Z ) - ~ m(X'Z))( m(P'Z)- xes ~ = m(P, Z) + ½ ~ m(X, Z)m(Z, Y ) - ~, re(P, Z)m(Z, X). YeS X¢Y
XeS
Hence
0<___zer ~ ez=d(P)-(s21) -
l(X, Y).
l(P, X) + + X~S
X, YeS X~ Y
Thus
d ( P ) > ( s 2 1 ) + ]~ I(P,X)-½ ~ i(x, Y). XeS
X, Y~S X¢ Y
[]
M.S. Montakhab / Finite pseudo-complements of quadrilaterals
105
We call a line y of a non-trivial (n ÷ 1)-regular linear space a//-line if y is of size r/--fl.
Lemma 2.2. The lines o f a pseudo-complement o f a quadrilateral S o f order n con9ist o f three l-lines, 6(n - 2) 2-lines and n 2 - 5n + 6 3-lines.
Proof. Let ai be the number of all (n - /)-lines of S. Then
ai = n2 + n - 3. i
Also, by simple counting methods,
iai = (n + 1)(n 2 - 3n + 3), i
i ( i - 1)ai = (n 2 - 3n + 3)(n 2 - 3n + 2), i
Hence ~i - n + 2 ) ( i - n + 3)ai = 6. i
~owever, S has at leastthree l-lines, and each of them contributes 2 to the left hand ;ide of the equality above. Thus a i = 0, i : / : n - 1 , n - 2 , n - 3 , and the lemma :ollows. [] It can easily be verified (see Mullin and Vanstone (1979)) that: Lemma 2.3. In a non-trivial (n + 1)-regular linear space with n 2 + n + 1 - f , f_> O, ines: (i) a fl-line is disjoint with exactly n(fl + 1 ) - f lines; (ii) a fl-line y and a ~-line z,y:#z, are mutually disjoint with exactly
n - 1)(1 - l Y, z[) + (/~ + l Y, zl)(,~ + l Y, zl) - f lines; (iii) a p o i n t P not in a fl-line y is contained in exactly fl + 1 lines disjoint with y. Using the same techniques as in the p r o o f of Theorem 3 of Mullin and Vanstone 1979), one can easily conclude that: ~ m m a 2 . 4 . / f n > 9 and y is a l-line o f a pseudo-complement o f a quadrilateral S ~f order n, then any line o f S disjoint with y is a 2-line.
It has been s h o w n by Vanstone (1973) that: ~emma 2.5. L e t S be a non-trivial (n + 1)-regular linear space. I f the number o f Joints is greater than n 2, then S is uniquely embeddable into a projective plane o f ~rder n.
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M.S. Montakhab / Finite pseudo-complements of quadrilaterals
3. New points and lines In this section, we will use the well known claw-clique technique to construct new points and new lines to serve in embedding of pseudo-complements of quadrilaterals. Suppose S is a non-trivial (n+ 1)-regular linear space with n E + n + 1 - f , f->0, lines. We define a simple graph G in which V(G) is the set of lines of S and two vertices are joined if and only if the corresponding lines of S are disjoint. Then we call G the line graph of S. We call a vertex P of G a fl-vertex if its corresponding line of S is a fl-line. Then by Lemma 2.3:
Lemma 3.1. (i) I f P is a B-vertex, then d(P)=n(fl+ 1 ) - f . (ii) I f P is a fl-vertex and Q is a t~-vertex with P ¢ Q , then ~n-l-f+flt~ l(P, Q) = ~.(fl + 1)(6 + 1) - f
if m ( P , Q ) = l , otherwise.
(iii) For a fl-vertex P, there exists a claw (P, S) o f order fl + 1. Henceforth, let G be the line-graph of a pseudo-complement of a quadrilateral of order n. We will construct maximal cliques by using non-extendable claws. A maximal clique corresponds to a maximal set of mutually disjoint lines. Since every line contains at least n - 3 points, no clique can have more than n vertices unless n _<6. The lines corresponding to a clique which consists of n - 3 3-vertices and three 2-vertices cover all the points of S. The same holds for the 'lines' of a clique consisting of one l-line and n - 2 2-lines. Such cliques are, therefore, maximal. Now, by Lemmas 2.1, 2.4 and 3.1, the following lemma is immediate:
Lemma 3.2. I f n> 14, then no claw o f order 2 at a l-vertex is extendable. Lemma 3.3. I f n> 14, then a l-vertex P is contained in exactly two maximal cliques H and K o f size n - 1. Each o f these contains n - 2 2-vertices, H N K = {P} and no 2-vertex in H is joined to any 2-vertex in K. Proof. Consider a claw (P, {Q,R}). By Lemma 3.1(iii), such a claw does exist and, by Lemma 3.2, it is maximal. The set of vertices joined to P can be partitioned into three subsets M, N and T where M consists of those vertices not joined to Q, N of those not joined to R, and T o f all those joined to both Q and R. Let H = M U {P} and K = N U {P}. Clearly, R e l l , Q e K and Ht')K= {P}. Now, by Lemma 3.1(ii),
I g l - 1 + Irl--IlVl- 1+ Irl. But, by Lemma 3.1(i), Igl+lNl+lrl=2n-4. Hence, I r l = o , I g l = [ N l = n - 2 , and thus H and K are maximal cliques of size n - 1. Suppose L is another maximal clique of size n - 1 containing P. Then by Lemma
M.S. Montakhab / Finite pseudo-complements of quadrilaterals
107
3. l(i, ii), L and H U K have no common 2-vertex. Therefore L is of size one, a contradiction. Now, by Lemma 3.1(ii), the result follows. [] In terms of structures, Lemma 3.3 can be stated as follows:
Lemma 3.4. I f n > 14, then a l-line y o f a pseudo-complement o f a quadrilateral o f order n is contained in exactly two classes H and K o f size n - 1 such that: (i) each class contains n - 2 2-lines; (ii) H N K = { y } ; (iii) two lines are in the same class i f and only i f they are disjoint.
By Lemmas 2.3(iii) and 3.4 it follows: Corollary 3.5. Let y and z be two distinct l-lines o f a pseudo-complement o f a quadrilateral o f order n. I f n > 14, then every 2-line disjoint with y is disjoint with exactly one 2-line in every class containing z.
Corollary 3.6. Let y be a l-line o f a pseudo-complement o f a quadrilateral o f order n. I f n > 14, then every 3-line is disjoint with exactly two lines in every class containing y.
Let S be a pseudo-complement of a quadrilateral of order n. We consider its linegraph G again. Then, by Lemmas 2.1, 3.1, 3.4 and Corollary 3.5:
Lemma 3.7. Let P be a 2-vertex o f G and n> 19. Then there exists a claw (P, {R1,R2,R3} ) in which R 1 is a 1-vertex and the others are 2-vertices. Furthermore, such a claw is not extendable.
Lemma 3.8. Let P be a 2-vertex o f G and n>21. Then G cannot have a claw (P, {R1,R2,R3,R4} ) in which R l is a 1-vertex, R 2 is a 2-vertex, and each o f the others is a 3-vertex.
Proof. Suppose such a claw does exist and let Hi, 1 _
But, by Lemma 3.1 and Corollaries 3.5, 3.6,
[H~NH21=O,
IHINH3[=IH, NH4I= I,
IH2NH~I-<7,
IH2NH4I <- 7,
IH3NHaI-< l l .
Hence n+l
i l
-`22
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M.S. Montakhab / Finite pseudo-complements of quadrilaterals
so that n-<21, a contradiction.
[]
Lemma 3.9. I f n > 23, then every 2-vertex P is contained in exactly three maximal cliques Ki, i= 1, 2, 3, o f size n - 1, where K l consists o f a unique 1-vertex and n - 2 2-vertices, and each o f the others consists o f three 2-vertices and n - 3 3-vertices. Furthermore, K i f3 Kj = {P }, 1 -
A = ~ m ( P , X ) ( 1 - m ( R I , X ) - m ( R E , X)). XeT
It is easily seen that the contribution of each vertex X to A is 1 if X e N a n d is nonpositive otherwise. Hence, by Lemma 3.1(i),(ii),
INI>-A=n-2. Thus, if/(3 is a maximal clique including {P} ON, then lgal ___n- 1. In a similar fashion, one can prove that R2 is contained in a maximal clique/(2 of size at least n-1. Suppose X E ( K i N K j ) \ {P}, l
Igi u gj I-< I(P, X ) + 2 < n + 3,
Igingil -
we have
2(n- 1)-< IK l + IK l-
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maximal cliques K i, 1 <_i <_4, o f size n. Each consists o f three 2-vertices and n - 3 3-vertices. Furthermore,
inKj--{P}, Proof. Let Q be a 1-vertex and H~ and H2 be the maximal cliques of size n - 1 containing Q. Then, by Corollary 3.6, P is joined to exactly two 2-vertices in both HI and HE. Thus, by Lemma 3.9, P is contained in at least four maximal cliques Ki, 1 < i < 4, of size n each of which consists of three 2-vertices and n - 3 3-vertices. Now, by using the same techniques as in Lemma 3.9, the result follows. [] In terms of structures, Lemmas 3.9 and 3.10 can be stated as follows:
Lemma 3.11. A 2-line y o f a pseudo-complement o f a quadrilateral o f order n, n>23, is contained in exactly three classes Ki, 1
2oroHary 3.13. Let S be a pseudo-complement o f a quadrilateral o f order n, n > 23. then every point o f S is contained in a unique line o f every class described in ~emmas 3.4, 3.11 and 3.12. ~,emma 3.14. A pseudo-complement S o f a quadrilateral o f order n, n > 23, has :xactly 4 n - 2 classes o f the type described in Lemmas 3.4, 3.11 and 3.12. 'roof. We count in two ways, the number of all ordered pairs (y, K) where y is an -line of S, i = 2, 3, and K is a class of size n which contains y. If a is the number ff classes of size n, then by Lemmas 3.11 and 3.12, 1 2 ( n - 2 ) + 4 ( n 2 - 5 n + 6 ) = n a , vhence ct = 4 n - 8 . The result follows by Lemma 3.4. []
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4. Embedding Let S be a pseudo-complement of a quadrilateral of order n, n > 23. If we add as new points all classes described in Lemmas 3.4, 3.11 and 3.12, to every line contained in them, then by Lemmas 3.13 and 3.14, we get a larger structure D whose dual D t is a non-trivial (n+ D-regular linear space with n 2 + n - 3 points and n2+ n + 1 lines. By 2.5, D t is uniquely embeddable into a projective plane of order n and the dual of a finite projective plane is also a finite projective plane. Thus:
Theorem. A pseudo-complement o f a quadrilateral of order n, n > 23, is uniquely embeddable into a projective plane o f order n.
Acknowledgement I would like to thank the referee for pointing out several errors and for other valuable suggestions.
References Hughes, D.R. and F.C. Piper (1973). Projective Planes, Springer, Berlin-Heidelberg-New York. Mullin R.C. and S.A. Vanstone (1979). Embedding the pseudo-complement of a quadrilateral in a finite projective plane, Ann. New York Acad. Sci. 319, 405--412. Vanstone, S.A. (1973). The extendibility of (r, 1)-designs, Proc. Third Manitoba Conference on Numerical Math., 409-418.