Bonferroni Inequalities and Negative Cycles in Large Complete Signed Graphs

Europ. J. Combinatorics (1996) 17 , 479 – 483

Bonferroni Inequalities and Negative Cycles in Large Complete Signed Graphs DRAGOS¸ POPESCU

AND

IOAN TOMESCU

In this paper the problem of characterizing extremal graphs Kn relatively to the number of negative p -cycles, when the number of negative edges is fixed, is solved for large n. This number can be expressed as an alternating sum for which the Bonferroni inequalities hold. Finally, the asymptotic value of the probability that a p -cycle of Kn is negative is found as n 5 ` , if the negative edges induce a subgraph the components of which are paths or cycles. ÷ 1996 Academic Press Limited

1. INTRODUCTION A signed graph G based on F is an ordinary graph F with each edge marked as positive or negative. In this paper we shall consider only the case in which F is the complete graph Kn . A cycle of Kn is said to be negative if it contains an odd number of negative edges; otherwise, it is positive [2]. Let us denote by Kp,q the complete bipartite graph the partite sets of which contain p and q vertices respectively, by sK2 the graph on 2s vertices consisting of s vertex-disjoint edges, by G (Kn ; H ) the complete signed graph Kn the negative edges of which induce a subgraph isomorphic to H , and by C 2 p (G ) the number of negative p -cycles contained by the signed graph G . If signed (or ordinary) graphs G and H are isomorphic, we shall denote this by G > H. A signed graph is called balanced if each of its cycles is positive. It is easy to show that a signed graph G based on Kn is balanced iff G > G (Kn ; Kp ,q) , where p 1 q 5 n and p , q > 0 . Psychologists are sometimes interested in the smallest number d 5 d (G ) such that a signed graph G may be converted into a balanced graph by changing the signs of d edges. It is easy to see that, for a signed graph G having d (G ) 5 k , there exists a cocycle v of Kn such that the signed graph obtained from G by changing the signs of all edges of v has exactly k negative edges.

2. BONFERRONI INEQUALITIES

FOR THE

SYMMETRIC DIFFERENCE

It is well known that for s sets A1 , A2 , . . . , As , their symmetric difference, denoted

nsi51 Ai , is the set of elements in !si51 Ai that belong to an odd number of sets

A1 , . . . , As . The following property is an analogue of the inclusion – exclusion principle for the symmetric difference of s sets. LEMMA 1.

The following equality holds :

U n A U 5 O (22) s

s

i51

i

i 21

i 51

479 0195-6698 / 96 / 050479 1 05 $18.00 / 0

O U " A U.

K’h1 ,... ,s j uK u5i

p PK

p

÷ 1996 Academic Press Limited

480

D. Popescu and I. Tomescu

PROOF. By Jordan’s sieve formula, we deduce that

U n A U 5 O O (21) Ski D O U " A U s

s

i 51

i

i 2k

k >1 i 5k k odd

K ’h1 ,... ,s j uK u5i

pPK

p

i 2k

i 51

k >1 k odd

O (22) s

5

p PK

O O (21) Ski D O U " A U s

5

K ’h1 ,... ,s j uK u5i

i 21

i 51

p

O U " A U. p PK

K ’h1 ,... ,s j uK u5i

h

p

We hay e

LEMMA 2.

U n A U < O (22) 2t 11

s

i 51

i

i 21

i 51

U n A U > O (22) 2t

s

i 51

i

K ’h1 ,... ,s j uK u5i

p PK

p

O U" A U

i 21

i 51

O U" A U

K ’h1 ,... ,s j uK u5i

s 21 ; 2

s for ey ery 1 < t < . 2

p

p PK

for ey ery 0 < t <

The proof is similar to that of the Bonferroni inequalities for the inclusion – exclusion principle (see [3, 4]), since unsi5111 Ai u 5 unsi51 Ai u 1 uAs 11u 2 2 unsi51 (Ai > As11)u. h 3. EXTREMAL NUMBERS

OF

WITH A

NEGATIVE p -CYCLES IN LARGE COMPLETE SIGNED GRAPHS GIVEN NUMBER OF NEGATIVE EDGES

We need the following auxiliary result, the proof of which can be set out using standard methods [1]. LEMMA 3. Suppose that the y ertex -disjoint paths P1 , . . . , Ps containing together t 1 s y ertices are induced subgraphs of Kn . The number of p -cycles of Kn that contain all paths P1 , . . . , Ps is equal to 2s21( p 2 t 2 1)! ( n n22t 2p s).

THEOREM 1. Let G be a complete graph of order n containing s negatiy e edges . 2 (a) If n > max(s 1 1 , 2p 2 2) , then C 2 p (G ) > C p (G (Kn ; K1 ,s)) and the equality holds iff : G > G (Kn ; K1 ,s) for s ? 3 and p > 3 or s 5 3 and p 5 3; G > G (Kn ; K1 ,3) or G > G (Kn ; K 3) for s 5 3 amd p > 4 . (b) If n > 2s for p 5 3 and n > 2p 2 2 1 2( 3s ) for p > 4 , then C 2 p (G ) < 2 C p (G (Kn ; sK 2)) and the equality holds iff G > G (Kn ; sK 2). PROOF. (a) Suppose that the negative edges of G (Kn ; K1 ,s) are denoted by e1 , . . . , es and let Ai be the set of p -cycles of Kn containing edge ei for i 5 1 , . . . , s. Then

Un A U 5 O A 2 2 O A > A s n 22 s n23 5 S D( p 2 2)! S 2 2S D( p 2 3)! S D D. 1 n 2p 2 n2p

C2 p (G (Kn ; K 1 ,s)) 5

s

s

i 51

i

u

i 51

iu

u

1
i

ju

(1)

Bonferroni inequalities

481

If G > u G (Kn ; K1 ,s) , let f1 , . . . , fs be the negative edges of G and let Bi be the set of p -cycles of Kn that contain edge fi for i 5 1 , . . . , s. By the Bonferroni inequalities we deduce that C2 p (G ) 5

Un B U > O B 2 2 O s

s

i 51

i

u

i51

iu

i
uBi > Bj u .

(2)

24 If the edges fi and fj have no common vertex, then uBi > Bj u 5 2( p 2 3)! ( nn 2 p ) by n23 n 24 n23 Lemma 3; otherwise, uBi > Bj u 5 ( p 2 3)! ( n 2 p ) and 2( n 2 p ) , ( n 2 p ) is equivalent to n > 2p 2 2 . For s 5 1 the property is obvious and for s > 2 and s ? 3 , since G> u G (Kn ; K1 ,s) , there exist two edges fi and fj having no common extremity; hence 23 n22 uBi > Bj u , ( p 2 3)! ( nn 2 p ) . Since, for every i , uBi u 5 ( p 2 2)! ( n 2 p ) , from (1) and (2) 2 2 it follows that C p (G ) . C p (G (Kn ; K1 ,s)). If s 5 3 , G > u G (Kn ; K1,3) and any pair of edges among f1 , f2 and f3 have a common extremity, then G > G (Kn ; K3) and in this case we obtain that C 2 p (G ) > C2 ( G ( K ; K )) and the equality sign occurs if f p > 4 . p n 1 ,3 (b) By the Bonferroni inequalities we deduce, as above, that

C2 p (G (Kn ; sK 2)) >

S1sD( p 2 2)! Spn 22 22D 2 2 S2s D( p 2 3)! Spn 22 44D. 2

(3)

Suppose that G > u G (Kn ; sK 2). With the notation introduced above we find that

26 uBi > Bj > Bk u is equal to: (i) 22( p 2 4)! ( np 2 6 ) if Hi ,j ,k > 3K 2 , where Hi ,j ,k denotes the

5 subgraph induced by fi , fj and fk ; (ii) 2( p 2 4)! ( np 2 2 5 ) if Hi ,j ,k is isomorphic to a graph of order 5 consisting of a path of length 2 and an edge; (iii) 0 for p > 4 and 1 for p 5 3 if 4 Hi ,j ,k > K3; (iv) ( p 2 4)! ( np 2 2 4 ) if Hi ,j ,k is isomorphic to a path of length 3 ; (v) 0 if Hi ,j ,k > K1 ,3. Now the proof follows in a similar way as above, by considering the cases p > 4 and p 5 3 . h

It is clear that d (G (Kn ; K1,s)) 5 d (G (Kn ; sK 2)) 5 s for every n > 2s. Hence, for large n , from the Theorem 1 we can deduce the structure of signed graphs based on Kn such that d (G ) 5 s having a minimum (resp. maximum) number of negative p -cycles. 4. NEGATIVE p -CYCLES

IN

COMPLETE SIGNED GRAPHS THE NEGATIVE EDGES INDUCE PATHS OR CYCLES

OF

WHICH

LEMMA 4. Let k and i be natural fixed numbers and let G be a graph with s edges such that d (x ) < k for ey ery x P V (G ). If Mi (G ) denotes the number of matchings of G containing i edges , the following equality holds : lim

s 5`

Mi (G ) 1 5 . si i!

PROOF. Since the number of ordered selections of i pairwise non-adjacent edges of G is greater than or equal to s (s 2 (2k 2 1)) ? ? ? (s 2 (i 2 1)(2k 2 1)) , it follows that s (s 2 (2k 2 1)) ? ? ? (s 2 (i 2 1)(2k 2 1)) / i ! < Mi (G ) < ( si ) and the result follows. h THEOREM 2. Let G 5 G (Kn ; H ) be a complete signed graph such that the negatiy e edges span a subgraph H containing s edges. If limn 5` s / n 5 l , limn 5` p / n 5 m and all components of H are paths or cycles , then 2pC 2 p (G ) 5 1–2 (1 2 e24lm ). n 5` (n )p lim

482

D. Popescu and I. Tomescu

PROOF. Suppose that the negative edges of G are e1 , . . . , es and let Ai be the set of p -cycles of Kn containing ei , for i 5 1 , . . . , s. We can write: C2 p (G ) 5

O (22) s

O U " A U.

i 21

i 51

t PK

K ’h1 ,... ,s j uK u5i

(4)

t

We shall prove that, for any fixed i , we have

O U " A UY(n) 5 (2li!m ) . i

lim 2p

n 5`

t

t PK

K ’h1 ,... ,s j uK u5i

(5)

p

Suppose that the s negative edges of G induce r components C1 , . . . , Cr that are paths or cycles, having a1 , . . . , ar vertices, respectively. The number of the selections of i edges from the set of the edges of a path P of length a 2 1 , such that these i edges 1 a 2i generate exactly j connected components on P , is equal to ( ij 2 2 1 )( j ) [5] and the a i 21 a 2i 21 – number of these selections for a cycle C of length a is equal to j ( j 2 1 )( j 2 1 ) [1]. By Lemma 3, we deduce that lim 2p

n 5`

O U " A UY(n) 5 lim 2pM(n, p, i, a , . . . , a )/(n) ,

K ’h1 ,... ,s j uK u5i

t

t PK

where

p

1

n 5`

r

p

O O P a(a , i , j )2 n 2i 2j 3S D 5O j n2p r

M (n , p , i , a1 , . . . , ar ) 5 ( p 2 i 2 1)!

i11???1ir 5i j1 ,... ,jr t 51 ,j

a (at , it , jt ) 5

a (at , it , jt ) 5

t

j 21

t

Sji 22 11DSa 2j i D t

t

t

t

if Ct is a path and

t

r

t 51

and

t

t

S DSa 2j 2i 21 1D

at it 2 1 jt jt 2 1

if Ct is a cycle. But a (at , it , jt ) <

t

t

t

Sji 22 11Ds , t

jt

t

which implies that

P a(a , i , j ) < s P Sji 22 11D. s

s

t

j

t 51

t

t

t

t51

t

Let N (n , p , s , i , j ) 5 2p ( p 2 i 2 1)! s j2j 21

Spn 22 ii 22 jjDY(n)

p

5 2js j ( p j 1 O( p j 21)) / (n i1j 1 O(n i 1j 21)). For j , i we have limn 5` N (n , p , s , i , j ) 5 0 . If j 5 i , or jk 5 ik for k 5 1 , . . . , r , the ik negative edges in Ck are pairwise non-adjacent for every k 5 1 , . . . , r , i.e. they are the edges of a matching of cardinality i of H. By Lemma 4 we obtain that Mi (H ) 1 5 s 5` si i! lim

Bonferroni inequalities

483

and if jk 5 ik for k 5 1 , . . . , r , then the corresponding part of the sum 2 2i M (n , p , i , a1 , . . . , ar ) is equal to ( p 2 i 2 1)! Mi (H )2i 21( np 2 2i ). It follows that 1 (2l m )i lim 2pM (n , p , i , a1 , . . . , ar ) / (n )p 5 lim N (n , p , s , i , i ) 5 n 5` i ! n5` i! which proves (5). Since the alternating sum (4) satisfies the Bonferroni inequalities, it follows, by standard techniques, that ` 2pC 2 (2l m )i 1 p (G ) h lim 5 (22)i21 5 –2 (1 2 e24lm). n 5` (n )p i ! i 51

O

Note added in proof : The authors proved that the conclusion of the Theorem 2 holds also if we suppose that the degrees of vertices of H are bounded above by an absolute constant C. REFERENCES 1. G. Baro´ ti, On the number of cetain Hamilton circuits of a complete graph, Period. Math. Hung. , 3 (1973), 135 – 139. 2. F. Harary, On the notion of balance of a signed graph, Mich. Math. J. , 2 (1953), 143 – 146. 3. L. Lova´ sz, Combinatorial Problems and Exercises , Akade´ miai Kiado´ , Budapest, 1979. 4. I. Tomescu, Problems in Combinatorics and Graph Theory , John Wiley, New York, 1985. 5. I. Tomescu, On the number of paths and cycles for almost all graphs and digraphs, Combinatorica , 6(1) (1986), 73 – 79. Receiy ed 9 July 1990 and accepted 20 June 1995 D. POPESCU AND I. TOMESCU Department of Mathematics , Uniy ersity of Bucharest , Str. Academiei , 14 , R-70109 Bucharest , Romania