ON GRAPHS THAT DO NOT CONTAIN A THOMSEN GRAPH
W. G. Brown
(received F e b r u a r y 7, 1966)
1.
A Thomsen graph [2, p . 22] consists of six v e r t i c e s
partitioned into two c l a s s e s of t h r e e each, with every v e r t e x in
one c l a s s connected to every v e r t e x in the other; it is the graph
of the "gas, water, and e l e c t r i c i t y " p r o b l e m [1, p . 206]. (All
graphs considered in this paper will be undirected, having
neither loops nor multiple e d g e s . )
We define g(n) to be the l a r g e s t integer m for which
t h e r e exists a graph of n v e r t i c e s and m - 1 edges containing
no Thomsen graph; (it may, however, contain a subdivision of a
Thomsen graph). It has been shown by Kôvari, Sds, and Turân
[7] that
- , - 1 / 3 5/3
tA A\
t \^
3n + 2
n
(1.1)
g(n) <
.
This has been improved by Znam [8]; but his r e s u l t still yields
the s a m e r e s u l t in the limit, viz.
lim sup
n -*• oo
n
g(n) <^ 2
Kovari et a l . [7] and E r d o s [5] have conjectured that
(1.2)
g(n)> c n 5 / 3
for some positive constant c.
conjecture c o r r e c t .
In this paper we prove that
The author wishes to acknowledge s e v e r a l enlightening
conversations with his colleagues, D r . R. Westwick and
D r . W. McWorter, during the course of this r e s e a r c h .
Canad. Math. Bull. vol. 9, no. 3, 1966.
281
2.
A lower bound for
g(n).
Let p be an odd p r i m e . We construct a graph G whose
3
v e r t i c e s a r e the p points of the affine geometry EG(3,p), i . e .
ordered t r i p l e s x = (x , x , x ) of elements of GF(p).
Define
S(x) to be the set of points y of EG(3,p)
for
which
£
(2.1)
(x.-y.)
1-1
where a is a fixed element of GF(p) chosen to be a n o n - z e r o
quadratic r e s i d u e if p = 3 (mod 4), and a quadratic n o n - r e s i d u e
o t h e r w i s e . Then, by a well known theorem of Lebesgue [3, p . 325]
2
the number of points in S(x) is p - p. We shall connect
v e r t i c e s x and y of G by an edge if and only if
y € S(x) ; or, what is equivalent,
x € S(y) .
3
2
This graph has p v e r t i c e s , each of valency p - p
5
4
(p - p )/2 edges.
thus
Suppose that G contains a Thomsen graph with v e r t i c e s
a, a*, a11; b, b 1 , b " and edges connecting each a with each b.
The points b, b», b11 m u s t lie in S(a) 0 S(a ! ) fl S(a n ). Thus
w = b, b ! , or b " a r e t h r e e solutions of the equations
!
2
2 (a. - w.)
i
n
= S (a. - w.)
2
= 2 (a . - w.) = a
i
i
l
2
l
l
hence also of the equations of the r a d i c a l planes of these s p h e r e s ,
viz.
i
l\
(2.2)
2A
=
2
'2
"2
a. - a
x
"2
where
282
l
;
a
a
1
- a
1
1
\ aa i. - a,
a
a
2
2
- a
- a
2
a
a
2
a„ - a
2
2
a
3
3
- a
3\
- a
3
3 " a3/
A is evidently singular. As the a's a r e distinct, the rank of A
is 1 or 2. Thus, if there exists a Thomsen graph as described,
either the a's or the b's a r e collinear. That this is impossible
is a consequence of the following l e m m a .
(2.3)
LEMMA.
No three points of S(x) a r e collinear.
Proof.
By a suitable translation we can a r r a n g e that
the line of points p a s s through the origin. Suppose
(2.4)
y = ra
(a 4= (0, 0 , 0 ) ; r ranges over
m e e t s S(x) in m o r e than two points.
yields the quadratic equation in T*
(2 a 2 ) r
l
2
- 2 (2 a.x. ) r
i l
GF(p)
Substituting (2.4) in (2.1)
+ 2 x 2 = or
l
which can have m o r e than two solutions only if
(2.5)
2 a.
(2.6)
2 a x =0
i i
(2.7)
2 x
l
= 0
Since a ==
j (0, 0, 0) we can assume without limiting generality
that a. 4= 0. Then
1
2
2
2
2
2 2
2
a or = a 2 x . = ( - a x - a x J
+ a (x + x ) by (2.6)
1
1
l
Z 2 3 3
1 2 3
= - (a x - a x )
by (2.5).
l
3 2
2 3;
This contradicts the choice of a since - 1 is a quadratic
residue if p = 1 (mod 4) and a quadratic n o n - r e s i d u e o t h e r w i s e .
We have thus shown that, for odd p r i m e s p,
283
g( P 3 )> £ V"-
(2.8)
F o r any 8 in the i n t e r v a l 0 < € < 1 there is an integer
N^
such that for all n > N- t h e r e exists a p r i m e p for which
n
> p > ( 1 -£ )
decreasing,
, .
,3.
n
[7, p . 371J.
5
p
4
5/3
- p
g(n)>g(p )> * - y * -
Hence, since g is non5/3
n
> —
Sn
4/3
+n
-
for all n > N. , from which (1.2) follows i m m e d i a t e l y .
over,
lim inf
n
-5/3
More-
g(n) > 1/2 . We cannot prove the
n
"°°
-5/3
existence of lim n
g(n) .
nr*oo
3.
Graphs without q u a d r a n g l e s .
Define f(n) to be the m a x i m u m integer m for which
t h e r e exists a graph G with n v e r t i c e s and m edges containing no q u a d r i l a t e r a l . It is proved in [7] that
-3/2
lim sup f(n) n
= 1/2. Using the following construction
n
"*°°
-3/2
it can be shown that lim f(n) n
= 1/2 . (The existence of
n-*oo
this limit, with a different value, was conjectured by E r d o s in
[5] and e l s e w h e r e . This construction has also been found independently by Rényi, M r s . Turân, and E r d ô s , and will appear in
a forthcoming paper . )
Construct for each odd p r i m e q a graph G as follows:
The v e r t i c e s of G a r e the points of PG(2, q) , i . e . the lines
through the origin in EG(3, q). Two v e r t i c e s
(x , x x ) = (fa , r a , r a ) , (x , x , x ) = (<ra , crb , <rb ) a r e
1 ^ 3
1 2 3
1 2 3
1 2 3
connected by an edge in G if and only if a and b a r e distince
points of EG(3, q) not collinear with the origin, and
a b + a b + a b = 0.
11
2 2
3 3
2
Then q
v e r t i c e s have valency q + 1, and q + 1 v e r t i c e s have
2
valency q. Thus G has q + q + l v e r t i c e s and
284
(g +q +1)
+ 0(q ) edges, but no q u a d r i l a t e r a l .
2
the proof of the latter to the r e a d e r .
We leave
REFERENCES
1.
C. B e r g e , Théorie des g r a p h e s . (Paris, 1958).
2.
H. S. M. Coxeter and W . O . J . Moser, G e n e r a t o r s and
Relations for D i s c r e t e Groups, (Springer Verlag, 1957).
3.
L . E . Dixkson, Theory of Numbers, Volume II, (Chelsea
Publishing Company, 1952).
4.
P . E r d ô s , E x t r e m a l p r o b l e m s in graph theory, in Theory
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5.
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6.
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7.
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Zarankiewicz. Coll. Math. 3, (1955), 50-57.
8.
S. Znam, Two improvements of a r e s u l t concerning a
p r o b l e m of K. Zarankiewicz, Colloq. Math. 13, (1965),
255-258.
University of B r i t i s h Columbia
285