On Graphs That Do Not Contain The Cube And Related
Problems
Rom Pinhasi
y
Miha Sharir
z
Otober 27, 2002
1
Introdution
Let Q denote the edge graph of the 3-dimensional ube (it has 8 verties and 12 edges).
The Turan number of Q is the maximum number of edges in a graph on n verties that does
not ontain Q. Bak in 1969, Erd}os and Simonovits [1℄ have shown that the Turan number
of Q is O(n8=5 ). In this paper, we provide an alternative simpler proof of this result. The
original proof in [1℄ was based on the assumption that the given graph G is regular, and
required a nontrivial tehnial lemma that redues the general ase to the ase of a regular
graph. Our proof does not need to assume that G is regular (and so does not use this
lemma), and follows a dierent approah that appears to be more powerful than the one
in [1℄. We demonstrate this by applying the tehnique to obtain Turan numbers for more
general graphs than Q, whih the method of [1℄ seems inapable of ahieving.
2
Graphs That Do Not Contain the 3-Dimensional Cube
Certain appliations of extremal graph theory involve bipartite graphs where the sizes of
the two vertex sets are dierent from eah other. For this reason, we state our main result
for the bipartite ase; the non-bipartite ase is then an immediate orollary.
A bipartite graph
Theorem 2.1
Q
has
(
O m
4=5
4=5
n
+ mn +
1=2
A B , with jAj = m, jB j = n, that does not ontain
G
nm1=2
) edges.
Let E denote the number of edges of G. We dene a onguration to be a 6-tuple
= (u; v; a; b; ; d) of distint verties of G, suh that a; ; v 2 A, u; b; d 2 B , and (v; u),
Proof:
Work
on this paper by Miha Sharir has been supported by NSF Grants CCR-97-32101 and CCR-00-
98246, by a joint grant from the U.S.{Israel Binational Siene Foundation, by a grant from the Israeli
Aademy of Sienes for a Center of Exellene in Geometri Computing at Tel Aviv University, and by the
Hermann Minkowski{MINERVA Center for Geometry at Tel Aviv University.
y Department
of Mathematis,
Massahusetts Institute of Tehnology,
Cambridge,
MA 02139,
USA;
roommath.mit.edu
z Shool
of Computer Siene, Tel Aviv University, Tel Aviv 69978, Israel, and Courant Institute of
Mathematial Sienes, New York University, New York, NY 10012, USA;
1
sharirs.tau.a.il
(a; u), (a; b), (v; b), (; u), (; d), (v; d) are all edges of G; see Figure 1. In other words, onsists of two C4 's (yles of length 4) with a ommon edge and with no other ommon
vertex. We say that a pair (a; b), (; d) of edges of G is tame if all four endpoints are distint,
and there exist at most two pairs (u; v) suh that (u; v; a; b; ; d) is a onguration. We say
that = (u; v; a; b; ; d) is a tame onguration if (a; b), (; d) form a tame pair.
b
a
v
u
d
Figure 1: A onguration in the proof of Theorem 2.1.
Let K be the set of all tame ongurations. A trivial upper bound for jK j is O(E 2 ),
beause the number of pairs (a; b); (; d) of edges of G is O(E 2 ), and eah of them, if tame,
gives rise to only two ongurations in K .
We next obtain a lower bound for jK j, as follows. Fix an edge e = (v; u) of G, with
v 2 A, u 2 B , and let Ge denote the graph whose verties are the neighbors of either u or v
in G, and whose edges are the edges of G that onnet pairs of these neighbors. Let Ae , Be ,
Ee denote the number of verties in A, of verties in B , and of edges of Ge , respetively.
Put Ve = Ae + Be .
We note that the (tame or untame) ongurations of the form (u; v; a; b; ; d), for the
xed pair of verties u; v, orrespond in a 1-1 manner to the vertex-disjoint pairs of edges
of Ge . For a vertex a of Ge , let Æe (a) denote the degree of a in Ge .
Lemma 2.2
4Ve 1).
The number of vertex-disjoint tame pairs of edges in
Ge
is at least
1
E
2 e
(Ee
Call any pair of edges, whih is not tame, a bad pair. We wish to bound the number
of bad pairs of edges in Ge . Suppose that (a; b), (; d) is a bad pair. Consider any other
pair (u0 ; v0 ), suh that (u0 ; v0 ; a; b; ; d) is also a onguration. If u 6= u0 and v 6= v0 then the
8-tuple (u; v; u0 ; v0 ; a; b; ; d) forms a forbidden opy of Q in G, ontrary to assumption; see
Figure 2. Hene, either all suh pairs (u0 ; v0 ) satisfy u0 = u, or all suh pairs satisfy v0 = v.
In the former ase we say that the pair (a; b), (; d) is a bad u-pair and in the latter ase we
say that the pair (a; b), (; d) is a bad v-pair.
By symmetry, it suÆes to bound the number of bad u-pairs, and twie this bound will
serve as a bound for the number of all bad pairs in Ge .
Let a; 2 A be two distint neighbors of u in Ge , and let b 2 B be a neighbor of a in
Ge (so b is a neighbor of v in G). There is at most one edge of Ge inident to (namely,
(; b)) that shares a vertex with (a; b).
Proof:
2
a
b
u
v
u0
v0
d
Figure 2: A double onguration forming a ube in G.
We laim that there is at most one vertex d 2 B (dierent from b) suh that (a; b),
(; d) is a bad u-pair. Indeed, assume to the ontrary that there are at least two suh
neighbors of , say d1 ; d2 . Sine (a; b), (; d1 ) is a bad u-pair, there exist at least two
distint verties v10 ; v100 2 A, dierent from v, suh that (u; v10 ; a; b; ; d1 ) and (u; v100 ; a; b; ; d1 )
are ongurations. Similarly, there exist at least two distint verties v20 ; v200 2 A, dierent
from v, suh that (u; v20 ; a; b; ; d2 ) and (u; v200 ; a; b; ; d2 ) are ongurations. Clearly, there
exist a pair of verties v1 2 fv10 ; v100 g, v2 2 fv20 ; v200 g, suh that v1 6= v2 . Then the eight
verties (v; v1 ; v2 ; ; u; b; d1 ; d2 ) form a forbidden opy of Q in G (in fat, it is a opy of Q
plus one main diagonal (u; v)); see Figure 3.
a
b
v
u
v1
v2
d1
d2
Figure 3: Two untame pairs f(a; b); (; d1 )g and f(a; b); (; d2 )g, forming a ube in G.
This ontradition shows that there exist at most two `bad' neighbors of in Ge , with
respet to the xed edge (a; b) of that graph: one of them is b, and at most one other vertex
d forms a bad u-pair (a; b), (; d) in Ge . Let N (u) (resp., N (v )) denote the set of neighbors
of u (resp., of v); thus Ae = jN (u)j, Be = jN (v)j. Then the number of bad u-pairs of edges
(a; b), (; d) is at most 2Ae Ee . Similarly, the number of bad v-pairs of edges is at most
2Be Ee . Therefore the total number of bad pairs in Ge is at most 2Ee (Ae + Be ) = 2Ee Ve .
It follows that the number Me of tame pairs of edges in Ge satises
Ee
Me 2Ee Ve = 21 Ee (Ee 4Ve 1);
2
as asserted. 2
Put
G1 = fe 2 G j Ee 8Ve g;
3
= fe 2 G j Ee < 8Ve g:
The total number jK j of tame ongurations thus satises
X
X Ee (Ee 4Ve 1)
X
jK j = Me Me 2
e2G
e2G
e2G
G2
1
1
P
P
1 X E 2 1 X E e2G Ee 2
e2G Ee
e
e
4 e2G
2 e2G
4E
2 :
P
P
Assume for the timePbeing that Pe2G Ee 12 e2G Ee ; the omplementary ase will be
treated later. Then e2G Ee 21 e2G Ee , and
1
1
2
1
P
jK j 2
e G
Ee
2
P
2
e G
Ee
16E
2 :
P
Note that e Ee = 4S , where S is the number of C4 's in G. Hene,
2
jK j 2SE 2S:
For eah pair of distint verties u; v of G (both in A or both in BP
), let Wu;v denote the
number of paths of length 2 that onnet u and v in G. Note that u=6 v2A Wu;v = W (A) ,
where W (A) is the number of paths of length
2 inG whose extreme verties are in A and
X
P
Wu;v
whose middle vertex is in B , and that
=
S . Similarly, u6=v2B Wu;v = W (B ) ,
2
6 2
u=v A
where W (B) is the number of paths of length
2 in G whose extreme verties are in B and
X Wu;v whose middle vertex is in A, and
2 = S . Then we an lower bound S by
6 2
u=v B
S
=
X Wu;v 2
2
u;v A
"
X
=
2
Wu;v
Finally, we have W
=
W
=
X
2
u B
Æ (u)
2
=
#
2
P
u;v 2A Wu;v
m
2
2
P
2
u;v A
2
2
Wu;v
= (W m)
2 2
(A) 2
X Æ (u)
2
u B
(A)
Wu;v
2
u;v A
(A)
2
2 , where Æ(u) is the degree of a vertex u in G. Hene,
X Æ (u)2
2
u B
2
() 2
Æ u
P
2
u B
()
Æ u
2n
2
P
2
( ) = 2E 2
2
n
u B
Æ u
E:
A
We next assume that W (A) 2 m2 , whih implies that S (W4(m)) . Finally, we assume
that E n, whih implies that W (A) En . Putting it all together, we obtain
( ) 2
2
2
2S + jK j = S2
E
!
(A) 4
= (Wm4 E) = E7
m4 n4
:
Combining this with the upper bound for jK j, whih also holds trivially for S , we obtain
E7
m4 n4
= O(E 2 );
4
W (A)
2
:
or E = O(m4=5 n4=5 ).
It remains to handle the ases that we have ignored so far. First, if E n then
learly E
P
satises the asserted bound. Next, suppose that W (A) 2 m2 . Sine W (A) = u2B Æ(2u) ,
we obtain
E
=
X
2
( ) n+
Æ u
u B
2
X
2
(Æ(u) 1) = O 4n +
!1=2
X Æ (u)
2
2
u B
u B; Æ (u) 1
3
n = 5 = O(mn = +n):
1 2
1 2
Note that by interhanging the roles of A and B , we may also assume that E m and that
W (B ) 2 n2 . Otherwise we get, as above, E = O (nm1=2 + m).
Finally, assume that neither of these inequalities hold but that
X
1 X E = 2S:
Ee >
2 e2G e
e2G
2
We have
X
2
Ee <
X
8
2
(Ae + Be) 8
2
(Ae + Be ) = 16(W (A) + W (B) );
e G
e G2
e G2
X
where the last equality is easily veried. Hene, S < 8(W (A) + W (B) ). Suppose, without
loss of generality, that W (B) W (A) , so S < 16W (A) . This implies
S
=
X Wu;v 2
u;v A
2
"
X
=
2
Wu;v
2
2
u;v A
In other words, we have
X
2
2
Wu;v
Wu;v
#
2
X
2
16Wu;v ;
u;v A
= O(W (A) );
u;v A
whih, using the Cauhy-Shwarz inequality, implies that
W (A)
=
X
2
Wu;v
u;v A
11=2
0
1=2
X
m
2 A
Wu;v
2
2
= O(m(W (A) )1=2 );
u;v A
2
= O(m2 ). Sine we assume that E n, we have W (A) En , implying that
E = O (mn1=2 ). The omplementary ase W (A) W (B ) yields, in a fully symmetri manner,
E = O (m1=2 n).
We have thus ompleted the proof of Theorem 2.1. 2
The general ase is now a straightforward orollary:
or
W (A)
Corollary 2.3
A graph with
n
verties that does not ontain
5
Q
has
(
O n8=5
) edges.
3
A Generalization
In this setion we generalize the method presented in Setion 2 to bound the Turan number
of more general families of graphs.
Let k m be positive integers. Let A1 ; A2 ; B1 ; B2 be four pairwise disjoint sets, so that
jA1 j = jB1 j = k and jA2 j = jB2 j = m. Dene Qk;m to be a bipartite graph whose set of
edges is (A1 B2 ) [ (A2 B1 ) [ M1 [ M2 , where Mi is a perfet bipartite mathing in
Ai Bi , for i = 1; 2. It is easily heked that Q2;2 = Q. The following theorem bounds
the Turan number of Qk;m, but only for graphs that satisfy an additional assumption (for
simpliity of presentation, we do not onsider the bipartite version of this ase):
Let 2 k m be positive integers, and let G be a graph on n verties whih
does not ontain a opy of Qk;m, and also does not ontain a opy of Kk+1;k+1 . Then G
4k
has at most O(n 2k+1 ) edges.
Theorem 3.1
Note rst that the number of edges of a graph that satises only the seond assumption of the theorem is O(n2 k ), and that this bound stritly dominates the bound
asserted in the theorem, so the rst assumption is non-redundant for the asserted bound.
Again, we assume without loss of generality that G is a bipartite graph. We dene a
onguration to be a (2k + 2)-tuple (u; v; a1 ; : : : ; ak ; b1 ; : : : ; bk ) of distint verties of G, so
that (u; v), and (ai ; bi ), (ai ; u), (bi ; v), for i = 1; : : : ; k, are all edges of G,
We say that a 2k-tuple (a1 ; : : : ; ak ; b1 ; : : : ; bk ) of distint verties is tame if (a1 ; b1 ); : : : ; (ak ; bk )
are all edges of G, and there are at most 2km edges (u; v) in G suh that (u; v; a1 ; : : : ; ak ; b1 ; : : : ; bk )
is a onguration. Every suh onguration will be alled a tame onguration. The proof
proeeds along the same lines as in the proof of Theorem 2.1, but is atually simpler beause of the seond assumption of the theorem. It proeeds by eetively showing that all
ongurations are tame.
Let E denote the number of edges of G, and let N denote the number of tame ongurations. An easy upper bound for N is 2kmE k = O(E k ).
We next obtain a lower bound for N . We x an edge (u; v) of G and dene Ge exatly
as in Setion 2, namely, its verties are the neighbors of u and the neighbors of v in G, and
its edges are the edges of G that onnet the neighbors of u to the neighbors of v. Dene,
as above, Ve and Ee to be the number of verties and edges of Ge , respetively.
We laim that any mathing of size k in Ge gives rise to a tame onguration. Indeed,
let (a1 ; b1 ); : : : ; (ak ; bk ) be suh a mathing, and onsider all the edges (ui ; vi ) (inluding
(u; v)), so that (ui ; vi ; a1 ; : : : ; ak ; b1 ; : : : ; bk ) is a onguration; note that the edges (aj ; bj )
are distint and vertex-disjoint.
Among the edges (ui ; vi ) one annot nd a mathing of size m, beause suh a mathing
would have indued a opy of Qk;m in G. On the other hand, there an be at most k indies i
with a ommon vi , and at most k indies with a ommon ui . Indeed, assume, without loss of
generality, that v1 = = vk+1 . Then A = fu1 ; : : : ; uk+1 g and B = fb1 ; : : : ; bk ; v1 g indue
a opy of Kk+1;k+1 in G, ontrary to assumption. Now take a maximum mathing among
the edges (ui ; vi ); its size is at most m 1, and we may write it as (u1 ; v1 ); : : : ; (uj ; vj ),
for some j m 1. Any other edge (ui ; vi ) must be inident to one of the 2j verties
Proof:
1
+1
6
, and eah of these verties is inident to at most (k 1) suh additional
edges, for a total of at most j +2j (k 1) (m 1)(2k 1) < 2km. In other words, we have
shown that, for any hoie of a mathing of size k from Ge , the orresponding onguration
(u; v; a1 ; : : : ; ak ; b1 ; : : : ; bk ) is tame.
The number of ways to pik k distint and vertex-disjoint edges from Ge is at least
(Ee (k 1)Ve )k ;
Ee (Ee Ve )(Ee 2Ve ) (Ee (k 1)Ve )
>
k!
k!
assuming that Ee (k 1)Ve .
The total number N of tame ongurations satises, using Holder's inequality,
u1 ; : : : ; u j ; v 1 ; : : : ; v j
N
k1!
X
j
e Ee >(k 1)Ve
(Ee (k 1)Ve )k P
j
e Ee >(k
P
k
(k 1)Ve )
1)Ve (Ee
k !E k 1
:
P
Arguing as before, one has e Ee = 4S , where S is the number of C4 's in G, and e Ve =
2W , where W is the number of paths of length 2 in G. Let us assume that S (k 1)W .
Then we have
X
j
e Ee >(k 1)Ve
(Ee (k 1)Ve ) X
e
and thus
N
(Ee (k 1)Ve ) = 4S 2(k 1)W 2S;
k(2!ESk)
k
1
:
Using the analysis in the previous setion we obtain, assuming E n and W 2 n2 , that
S = (E 4 =n4 ). Thus, N = (E 3k+1 =n4k ). Combining this with the upper bound O (E k ),
k
we get E = O(n k ).
The remaining ases E < n, W < 2 n2 , or S < (k 1)W , are analyzed in a manner
similar to that in Setion 2. (Reall that these ases yield the bound E = O(n3=2 ), whih
is dominated by the bound asserted in the theorem, provided that k 2.) 2
Remarks: (1) We do not know whether Theorem 3.1 also holds without the assumption
that G does not ontain Kk+1;k+1.
(2) The approah of [1℄ seems inapable of obtaining this bound.
4
2 +1
Referenes
[1℄ P. Erd}os and M. Simonovits, Some extremal problems in graph theory, Combinatorial Theory and Its Appliations 1 (Pro. Colloq. Balatonfured, 1969), North Holland,
Amsterdam, 1970, pp. 377{390.
7