Biclosure and bistability in a balanced bipartite graph

Biclosure and Bistability in a Balanced Bipartite Graph

Denise Amar LABRI, UNIVERSITf DE BORDEAUX I

351 COURS DE LA LIBERATION 33405 TALENCE CEDEX, FRANCE

Odile Favaron* L.R.I. URA 410 CNRS, BAT. 490,

UNIVERSITE PARIS-SUD 91405 ORSAY CEDEX, FRANCE

Pedro Magot MA THEMATICS DEPARTMENT

SCHOOL OF SCIENCE UNIVERSIDAD DE ORIENTE

CUMA NA, VENEZUELA

Oscar Ordazt MA THEMA TICS DEPARTMENT

FACULTY OF SCIENCE UNIVERSIDAD CENTRAL DE VENEZUELA

AP. 47567, CARACAS 1041-A, VENEZUELA

ABSTRACT

The k-biclosure of a balanced bipartite graph with color classes A and 5 is the graph obtained from G by recursively joining pairs of nonadjacent vertices respectively taken in A and 5 whose degree sum is at least k, until no such pair remains. A property P defined on all the balanced bipartite graphs of order 2n is k-bistable if whenever G + ab has property P and &(a) + &(b) z k then G itself has property P.

We present a synthesis of results involving, for some properties P, the bistability of P, the k-biclosure of G, the number of edges and the minimum degree. 0 1995 John Wiley & Sons, Inc.

*Research partially supported by PRC Math. Info. 'Research partially supported by the French PCP-Info. (CEFI-CONICIT).

Journal of Graph Theory, Vol. 20, No. 4, 51 3-529 (1 995) 0 1995 John Wiley & Sons, Inc. CCC 0364-9024/95/040513-I 7

514 JOURNAL OF GRAPH THEORY

I. INTRODUCTION

Let G be a balanced bipartite simple graph of order 2n and color classes A and B. The vertex set is V = A U B with (A( = (BI = n, and the size e(G) = (E l . If the two vertices a and b are not adjacent, we say that ab is a non-edge of G. We denote by dG(a) the degree of a in G, by 6 (resp. A ) the minimum (resp. maximum) degree of G and by d(x , y ) the distance between two vertices x and y . An xy-path means a path with endvertices x and y and Ck (resp. Pk) is a cycle (resp. a path) of order k.

The graph G is hamiltonian if it contains an elementary cycle of length 2n (hamiltonian cycle), traceable if it contains an elementary path of length 2n - 1 (hamiltonian path), hamilton-biconnected if for any two vertices a in A and b in B, there exists a hamiltonian ab-path. It is bipancyclic if it contains cycles of all even lengths and bipanconnected if for every pair of vertices a in A and b in B, there exist ab-paths of every odd length between d(a , b ) and 2n - 1. It is quasi-bipanconnected if for every pair of vertices a in A and b in B, there exist ab-paths of every odd length L between 1 and 2n - 1 except perhaps for one length, L = 1 if a and b are not adjacent or L = 3 if a and b are adjacent. Finally G is 2s-hamiltonian (resp. 2s-traceable, 2s-hamilton-biconnected) if for any balanced subset W of order 2s of V, the subgraph induced by V - W (and often denoted by G - W) is hamiltonian (resp. traceable, hamilton-biconnected). The bipartite independence number aHIp(G) is the order of a largest balanced independent subset of V. An s-factor of G is an s-regular spanning subgraph of G. We denote by c the complement of G with respect to K I A I , ~ B ~ , by G U H the disjoint union of G and H (in particular 2G = G U G ) and by G + H the graph obtained from G U H by adding all the edges between G fl A and H fl B and between G f l B and H fl A . These last definitions are valid even if G and H are not balanced.

For the other notation, we follow the terminology of [6] and we use notions of closure of a graph and stability of a property as defined in [5] .

1.1 The k-biclosure of a graph.

The k-biclosure BGk of G with respect to the complete bipartite graph K, , , , is the graph obtained from G by recursively joining by an edge every pair of non-adjacent vertices respectively belonging to the two classes and of degree sum at least k. The result does not depend on the order of addition of the edges. We always have: G = B C ~ A + I C . . . C BGk+, L BGk C ... C BG28 =

K, , , for 0 5 26 5 k 5 2A + 1 f 2n + 1, and BG2,-, = G. For 0 f k 5 2n - 1, let us consider the following conditions:

Q2(k) For every non-edge a b with a in A and b in B, dG(a) + dG(b) 2 k . Q I ( ~ ) BGk = K n , n .

BICLOSURE AND BlSTABlLlTY 515

Q3(k) 6 2 k/2. Q4(k) e 2 (n - 1)2 + k.

It is clear that

Q3 \

Q4 /” Q2 -+ Q i

and that Q3(k) and Q4(k) are not comparable.

1.2 The k-bistability of a property

Let P be a property defined on all the balanced bipartite graphs of order 2n. The property P is k-bistable if:

a in A, b in B and ab non-edge - & ( a ) + &(b) < k G does not satisfy P

G + a b satisfies P

Note that every k-bistable property is ( k + 1)-bistable and that every property of G is (2n - 1)-bistable. The bistubility b s ( P ) of a property P is the smallest integer k for which P is k-bistable.

The basic idea of Bondy and ChvBtal, particularly well adapted to the properties of hamiltonicity, is the following:

Theorem 1.2.1 [ 5 ] .

P satisfied by K,,n and to an integer k between 1 and 2n - 1.

If P is k-bistable and if BGk has P, then G has P.

Let us consider the following statements of theorems related to a property

To(P, k) P is k-bistable T,(P, k) If G satisfies Q,(k), then G has P (1 5 i 5 4)

These theorems are less and less strong since

Moreover, if T i ( P , k) is best possible for some property P, some i with 0 < i 5 3 or 4 and some k, and if T j ( P , k) if true for some j and k with 0 5 j < min(3, i}, then T,(P, k) is also best possible.


1.3 Synthesis of results

Many papers in the literature give sufficient conditions for a bipartite graph to have a given property P in terms of the number of edges or of the minimum degree of G. We think it is not in vain to take stock of them although Bondy and Chviital wrote in [ 5 ] , quoting from [7], that “this here.. .may lead to a fine old mess.”

For some properties, we survey these results from the angle of the biclosure and of the bistability, pointing out the strongest already proved theorems T, (0 5 i 5 3 or 4) and we complete or improve some of them. Besides Section I containingthe introduction, this paper consists of three main sections: Section 11, where we study the existence of cycles, Section 111, where the existence of paths is discussed and Section IV, where we deal with other properties.

The results are gathered in Table I with six entries. The first column corresponds to various properties and refers to the sections where they are studied. Each of the other columns corresponds, for each property P, to a theorem T, ( P , k ) as defined in 1.2. We give the smallest value of k for which T, ( P , k ) is established (we mark with bp if this value is best possible) and the reference of the results which are already known.

I I . EXISTENCE OF CYCLES

11.1 Property P: G is harniltonian

The first result on the topic is the following:

Theorem 2.1.1 (Bondy-Chviital [ 5 ] ) . ( n + 1)-bistable.

The property P: G is lzanziltonian is

The theorems T, (P ,n + l), 2 5 i 5 4, have been known since 1963

This property admits three kinds of generalization: (Moon-Moser [9]) and are best possible.

G is 2s-hamiltonian, G contains a cycle Czs for other values of s 2 2, and G is bipancyclic.

11.2 Property P: G is 2s-hamiltonian.

Theorem 2.2.1. bistable.

Proof.

The property P: G is 2s-hamiltonian is ( n + s + 1)-

Suppose that for some balanced subset W of order 2s of V, the graph F = G - W is not hamiltonian and that there exists a non-edge ab of G with a in A and b in B such that G + a b is 2s-hamiltonian.

TAB

LE I

To(P

, k)

Ti (P

. k)

Tz (P

, k)

T3 ( P, k)

T4

( P. k)

e(

G) 2

(n

-

1)’

+ k

n +

s +

1 ib

p)

n +

s +

1 (b

p)

n +

s +

1 (b

p)

s ==

0:

Moo

n-M

oser

19)

s

= 0

: M

oon-

Mos

er 1

91

s =

0:

Moo

n-M

oser

191

PR 0

P E R

TY

Bis

tabi

lity

2 k

BG

k =

Kn,

n &

(a)

+ &

(b)

2 k

6 2 k

/2

2 s-

Ham

ilton

ian

[11.

1-11

.2)

Cyc

le C

zS 11

1.31

2n - s

+ 1

(bp

) n

+ 2

n+

l

n+

l

n+

l

Bip

ancy

clic

111

.41

11 +

2 n

+ 1

(bp)

n +

1 (

bp)

n +

1 !b

p)

n +

s ib

p)

2s-T

race

able

llll

.11

Pat

h Pt

[111

.21

n ~

It/21

+ 1

(bp

) n

+l

n

+ 1

n

+l

n

2 s-

Ham

ilton

ian-

n

+ s i

1 (b

p)

bico

nnec

ted

[I 11.3

1 n

+ 1

(bp

) B

ipan

conn

ecte

d [11

1.41

Qua

si-B

ipan

conn

ecte

d n

+ 2

(bp)

n

+ 2

(bp)

n

+2

a5

ip 5

2s

IlV.1

I s-

Fact

or (

s 2 2

) 1I

V.21

n

+ 2s

- 3

(bp

for

n+

2s

-3

n+

2s

-3

n

+2

s-

3

n+

2s

-3

1-Fa

ctor

[IV

.ZI

n BF

os)

n (b

p)

n (b

p)

n (b

p)

n (

bp)

n +

s +

1 (b

p)

s =

0:

Bon

dy-C

hvat

al 1

51

D +

s i-

I (b

p)

Bag

ga-V

arrn

a 13

1 M

itche

m-S

mei

chel

I81

S

mei

chel

-Mitc

hem

I1 01

n +

s +

1 (b

p)

n +

s +

2 (b

p)

n +

s +

1 (b

p)

n +

s + 2

ibp)

n +

2 (b

p)

n +

s +

1 (b

p)

n +

s +

2 (b

p)

n +

s +

1 ib

p)

n +

s + 2

(bp)

s

= 0

: Bag

ga-V

arm

a [3

1

[111.4

1 2

n - 2

s -

1 (b

p)

2n

- 2

s - 1

ibp)

2n

- 2s - 1

(bp

i 2n

~ 2s

~ 1

(bp)

2n

- (

s -t

1)’

(bp)


This implies F + ab is hamiltonian, thus a and b are in V - W and by Theorem 2.1.1, & ( a ) + &(b) 5 d,(a) + d F ( b ) + 2s < (n -- s) + 1 + 2 s = n + s + 1 . I

The graphs G l ( s ) = K S , $ + 2K(n-s)/2,(n-\)/2 if n + s is even and G ~ ( s ) =

K\ ,s + (K(n-~+1)/2,(n-s+1)/2 U K ( , ~ - \ - I ) , ~ , ( ~ - ~ - I ) / ~ ) if n + s is odd show that T, ( P , n + s + 1) are best possible for 0 I i 5 3.

The graph G 3 ( s ) obtained from Kn,,, by deleting n - s - 1 edges incident to the same vertex shows that T4(P,n + s + 1) is best possible.

11.3 Property P: G contains the cycle C&, 2 I SI n- 1.

Theorem 2.3.1. The property P: G contains C2s is (2n - s + 1)-bistable.

If G + ab contains a cycle C2r, but not G , then G contains a path a l h l ... a r b r with a1 = a and bs = b. The subgraph H induced in G by {a, , b,}l,,,, is not hamiltonian, whereas H + ab is hamiltonian. By Theorem 2.1.1, d H ( a ) + d H ( b ) < s + 1 and thus & ( a ) -i- dc(b) < 2(n - s) + s + I = 2n - s + 1. I

The graph G4(s) = K l , n - \ U Ks- l ,s- I U Kn-\, I plus one edge between Kl,,r-s n A and Ks-l,s-l fl B and all the edges between K s - l , y - ~ n A and Kn-s, I n B , shows that To(P, 2n - s + 1) is best possible. This is no more true for TI ( P , 2n - s + 1) when i > 0 as shown in the following paragraph where Tl(P,n + 2) and T , ( P , n + 1) for 2 5 i 5 4 are proved for every value of s. These new theorems themselves are probably not best possible. In the particular case of a C4, studied as a K2,2 as part of Zarankiewicz's problem, it is known that T4(P, i ( n + nd-)) is true [4].

Proof.

11.4. Property P: G is bipancyclic

The property P: G is harniltonian is (n + 1)-bistable. However the property P: G is bipancyclic is not ( n + 1)-bistable. For n odd, the graph ( 2 5 of order 2n defined by the cycle al bl a2b2 . . . a,, b,, a1 with additional edges a l 6, for j odd , 3 5 j 5 n - 2, and a,b2 for iodd, 5 5 i 5 n G5 by, has no cycles of length 4, while the graph G5 + albz is bipancyclic and d ( a l ) + d(b2) =

n + 1. The value of the bistability property P: G is bipancyclic is unknown. Nevertheless, we shall show that Theorems T I ( P , n + 2) and T,(P,n + l), for 2 5 i I 4, hold.

Theorem 2.4.1. If BGn+2 = Kn,n then G is bipancyclic.

In order to prove T, (P ,n + 2) (i.e., Theorem 2.4.1) we shall use the following results:

Theorem 2.4.2 [I]. Let G be a bipartite graph of order 2n and let C be a hamiltonian cycle in G. If there exist two vertices X I , x2 the distance between

BICLOSURE AND BlSTABlLlTY 51 9

which is 2 on C, such that &(XI ) + d ~ ( x 2 ) 1 n + 1, then G is bipancyclic except in the following two cases:

(1) n is odd, N ( x l ) = { y j , j odd, 1 5 j I n} and N(x2) = { y j , j even, 2 5 j 5 n - 1, and j = l} G may not contain cycles of length 4.

(2) n is odd, n = 2s - 1, N(x1) = N ( x 2 ) = { y l , . . . , y d } U { y s f d , . . . , y n } with d 5 s - 1. G may not contain cycles of length 2s.

Lemma 2.4.3. Let G be a balanced bipartite graph with degree sequences &(al) 2 &(a2) 3 . . . 2 &(a,) and dG(bl) 2 dc(b2) 2 . . . 2 dc(b,), and such that BGn+2 = K,,n. Then either & ( a , ) + &(a2) 2 n + 1, or dc(bl) + dc(b2) 2 II + 1.

Proof. Since BG,+2 + K,,,, d c ( a l ) + & ( b l ) L n + 2. If dG(a1) + d ~ ( a 2 ) 5 n and dc(b l ) + dc(b2) 5 n, then & ( a , ) + dG(a2) 5 n - 2. In the process to generate BGn+2. We by, we can first add only edges al 6 , and a ,b l with i 2 2, which are not in G. But then the new degrees of a, , for i 2 2, and b,, for j 2 2, are not big enough to add edges between them, a contradiction to the hypothesis BG,+2 = K, , , , . I

Proof of Theorem 2.4.1. We can suppose d G ( a 0 + dG(a2) 2 n + 1. Then al and a2 have a common neighbor b in B. Let H = (A, B, E ( H ) ) be the graph with E ( H ) = E(G) - {a ,b E E(G): i # 1,2}. Then d H ( u ) 2

&(a) - 1 and dH(b,) = dc(b,) for b, # b. It is easy to see that BH,+I =

K,,,, and then H has a hamiltonian cycle C which contains the path a l ba2. And by Theorem 2.4.2, G is bipancyclic.

The Theorem T2(P,n + 1) is also mentioned in [ 3 ] . It can be seen as a consequence of the theorem of Moon and Moser [9] and the following Theorem 2.4.4 of Schmeichel and Mitchem [ 101 by a counting argument on the number of edges.

Theorem 2.4.4. e (G) > n2/2 then G is bipancyclic.

I

Let G be a hamiltonian bipartite graph on 2n vertices. If

Theorems T3(P, n + 1) and T4(P , n + I ) are immediate consequences of

The graphs GI(()) for n even, Gz(0) for n odd, show that 7'2(P,n + 1)

The graph G7(0) shows that T 4 ( P , n + 1) is best possible. Motivated by Theorems T I ( P , n + 2)and T,(P,n + 1) for 2 5 i 5 4, it

T2(P,n + 1).

and T 3 ( P , n + 1) are best possible.

is natural to ask if the following conjecture is true:

Conjecture 2.4.5. If BG,+I = K, , , , then G is bipancyclic.

The following result strengthens our conviction. We need some definitions:


Definitions 2.4.6. We denote G = Go, and for k 2 1, Gk is the graph obtained from GkPl by joining all pairs of nonadjacent vertices a E A , b E B such that dkel(a) + dkPl(b) 2 IZ + 1 (where dl(x) denotes the degree of x in G I ) . Step k denotes the process of obtaining the graph Gk from the graph G k - l . If N is the least integer such that GN = K , , , , we say that BG,+, = Kn," in N steps.

Theorem 2.4.7. If G is a balanced bipartite graph of order 2n such that BG,+I = Kn," in N steps, then the size of G is more than n(n + N ) / ( N + 1).

Proof. For 1 5 k 5 N , if ( x i , y j ) E E(Gk)\E(Gk-i) then dk-l(Xi) + d k P l ( y j ) 2 n + 1. Hence

By summing from k = 1 to k = N , and using the equalities d ~ ( x i ) =

d N ( y J ) = n and e ( G N ) = n2 , we obtain:


Applying the Cauchy-Schwarz inequality:

1 n3 2 ( n + 1)(n2 - e (G) ) +

Then by applying again the Cauchy-Schwarz inequality we obtain that

----(I + i) - e ( G ) ( n + 1 + + n 2 + - n3 5 0 N N

e(G>I2 n

Therefore e(G) 2 [n(n + N ) ] / ( N + 1).

Remark 2.4.8. The Cauchy-Schwarz inequality can be an equality only when the graph is regular. But the condition BG,,+, = K",,, implies that at least two independent vertices x, and y , satisfy: dc(x,) + &( y , ) 2 n + 1 . So if N 2 2, the inequality is strict.

Remark 2.4.9. In the case N = 2, we can get more precise inequalities. As G2 = Kn." and by the previous remark,

I

Then

n n

i.e. d:(xi) + z d:( y;) I (3n + 1) (e(G1)) - n2(n + 1). ( 3 ) i = 1 j = 1


From (2) and ( 3 ) , 4(n + l)e(G) 2 (n + 3)e(G1) + n2(n + 1) and by (1) we have e(G) > [n(3n + 3)]/8. As a consequence, if [n(3n + 3)]/8 2

[n(n + 3)]/3 or, equivalently, if n 2 15, we can apply the following theorem:

Theorem 2.4.10 [2]. Let G be a hamiltonian bipartite graph of order 212, with more than [n(n + 3)]/3 edges, then G is bipancyclic. We now obtain:

Theorem 2.4.11 If BG,+l = Kn,n in at most 2 steps, for n -2 15, G is bipanc yclic.

111. EXISTENCE OF PATHS.

The results in this section differ from the corresponding ones on general graphs on the following two points: -The bistabilities of the properties P: G is hamiltonian and P: G is traceable are both n + 1, whereas the stabilities of these properties are respectively n and n - 1 in general graphs. -The bistability of the property P: G contains a path P , depends on t, whereas the stability of this property is n - 1 for every t in general graphs PI.

111.1 Property P: G is 2s-traceable.

Theorem 3.1.1. For s 2 0, the property P: G is 2s-traceable is ( n + s + 1)-bistable.

Proof. Let us suppose that G is not 2s-traceable, that is there exists a balanced 2s-subset W of V such that F = G - W is not traceable, but that for a non-edge ab of G with a in A and b in B, G + ab is 2s-traceable. This implies that F + ab is traceable, a and b belong to V - W and &(a) + &(b) 5 d ~ ( a ) + d ~ ( b ) + 2s. In order to prove that &(u) + &(b) < n + s + 1, it is sufficient to prove that d ~ ( a ) + dF(b) < (n - s) + I and thus that the property P: G is truceable is ( n + 1)-bistable. This is done later, as a particular case of Theorem 3.2.1 when t = 2n.

The same graphs GI($) and G ~ ( s ) as in Section 11.2 show that T , ( P , n + s + 1) are best possible for 0 2 i 5 3 . However Td(P,n + s + 1) is not and the following theorem improves it to T4(P,n + s).

Theorem 3.1.2.

If W is a balanced subset of order 2s of V, the size of F = G - W satisfies e ( F ) 2 e(G) - s2 - 2s(n - s) 2 ( n - s - 1)2 + n - s. In order to prove that F is traceable, it is thus sufficient to prove Theorem 3.1.2. for s = 0. We suppose therefore e(G) 2 (n - 1)* + n . If G is not complete

I

If e(G) 2 (n - 1)2 + n + s, then G is 2s-traceable.

Proof.


bipartite, let ab be a non-edge with a in A and b in B. The graph H = G + ab satisfies e ( H ) 2 ( n - 1)2 + n + 1 and, by paragraph 11.1, contains a hamiltonian cycle which provides a hamiltonian path in G.

The graph G ~ ( s ) = Ks,s + (Kn-s -~ ,n - . y U K I , ~ ) , with ( n - 1)’ + n + s - 1 edges, shows that T4(P,n + s) is best possible.

111.2 Property P: G contains a path Pi

Theorem 3.2.1. For t 5 2n, the property P: G contains a path PI is (2n - [t /2] + 1)-bistable.

Proof. Suppose that for a non-edge ab of G with a in A and b in B, G contains no path Pr but that G + a b contains a path ~ 1 x 2 . . . x,. Then ab is an edge of this path and, if t is even, x l x r is a non-edge of G. Let H be the subgraph of G induced by {x , ; 1 5 i 5 t } .

If t = 2k, the balanced graph Q = H + x l x t has no hamiltonian cycle, which would provide a PI in H , but x i x 2 ~ ~ ~ x , x I is a hamiltonian cycle of Q + ab. By Theorem 2.1.1, d ~ ( a ) + de(b ) < k + 1 and thus &(a) + &(b) < 2(n - k ) + k + 1 = 2n - k + 1 . If t = 2k + 1 , the balanced graph Q obtained from H by adding a new vertex xo adjacent to xi, xj, x5,. . . , x, contains no hamiltonian cycle but ~ 0 ~ ~ x 2 . . . xrxo is a hamiltonian cycle of Q + ub. By Theorem 2.1.1, d e ( a ) + d e ( b ) < ( t + 1)/2 + 1 . Therefore & ( a ) + &(b) < ( t + 1)/2 and thus dG(a)dG(b) < (n - ( t + 1)/2) + (n - ( t - 1)/2) + ( t + 1)/2 = 2n - [t/21 + 1.

K2,n-k+l if t = 2k + 1 show that To(P,2n - [t/2] + 1) is best possible. However, T,(P,2n - [ t /2 ] + 1) is not best possible for i 2 I and t < 2n. The theorems T,(P,n + 1) for 1 5 i 5 3 and T4(P,n) are obvious consequences of the corresponding results on the existence of a hamiltonian path.

The graphs Gl(0) if n is even and G2 = K ( n - i ) / 2 , ( n + i ) / 2 U K ( n + 1 ) / 2 , ( n - 1 ) / 2 if n is odd contain no path P,+l and show that for t 2 n + 1, the theorems T,(P,n + 1), 1 5 i 5 3 are best possible. The graph Gj(0) shows that for t = 2n, the theorem T,(P,n) is best possible.

I

The graphs G7 = K,,-i,k- 1 U K I , ~ - ~ + if t = 2k and Gx = Kn-2,k- 1 U

111.3 Property P: G is 2s-hami/ton-biconnected

Theorem 3.3.1. is (n + s + 2)-bistable.

For s 2 0, the property P: G is 2s-hamilton-biconnected

Proof. Let us suppose that G is not 2s-hamilton-biconnected, that is there exists a balanced subset W of order 2s of V and two vertices x in


A - W and y in B - W such that the subgraph F induced by V - W contains no xy-hamiltonian path. Suppose also that for a non-edge ah of G with a in A and h in B, G + ab is 2s-hamilton-biconnected. This implies in particular that F + ab contains a xy-hamiltonian path and thus a E A - W and b E B - W . The graph H obtained from F by adding two new vertices u, u and the path xuuy, is not hamiltonian but H + ab is hamiltonian. By Theorem 2.1.1, d n ( a ) + d ~ ( b ) < IHl + 1 = n - s + 2. But d H ( a ) 2 dF(a) and d ~ ( b ) 2 d F ( b ) . Moreover &(a) 9 d ~ ( a ) + s and dG(b) 9 d F ( b ) + s. Therefore & ( a ) + AG(b) < n + s + 2. I

For s = 0, the Theorem T I ( P , n + 2) was already proved in [3]. The graphs G l ( s + I ) if n + s is odd and G2(s + 1 ) if n + s is even show that T,(P,n + s + 2) are best possible for 0 f i f 3 . But the following theorem improves T4(P,n + s + 2) to T4(P ,n + s + 1).

Theorem 3.3.2. biconnected.

If e ( G ) 2 ( n - + n + s + 1 then G is 2s-hamilton-

Proof. By an argument similar to that which was used in Theorem 3.1.2, it is sufficient to prove Theorem 3.3.2 for s = 0. We proceed by induction on n. The property is clear for n = 2 since then G = K2,2. Suppose it is true until order 2(n - 1) and let G be a balanced bipartite graph of order 2n with e ( G ) 2 ( n - 1)' + n + 1. By Theorem T4(P,n + s + 2), it is sufficient to consider e ( G ) = (n - + n + I , which implies 6 2 2. Let x in A and y in B be two vertices of G.

If x and y are nonadjacent, the graph H = G + xy, which satisfies e ( H ) = (n - 1)' + n + 2, admits an xy-hamiltonian path by T4(P,n + s + 2), which is also an xy-hamiltonian path in G.

If x and y are adjacent, the graph Q = G - {x,y} satisfies e ( Q ) =

If &(x) + &(y) = 2n, then e ( Q ) = ( n - 2)* + ( n - 1) + 1 and by Theorem 3.1.2, Q admits a hamiltonian path which provides an xy- hamiltonian path in G.

If &(x) + & ( y ) 5 2n - 1, then e ( Q ) 2 (n - 2)' + ( n - 1) + 1 and Q is hamilton-biconnected by the induction hypothesis. Let x' and y ' be respective neighbors of x and y in Q (recall that 6 ( G ) 2 2). An x ' y l - hamiltonian path of Q provides an xy-hamiltonian path in G.

The graph Gg(s), obtained from K7,s + ( K n - s - ~ , n - 5 - ~ U K I J ) by adding all the edges between K I , ~ n A and Kn-s-l ,n-7-l f l B , shows that T4(P,n + s + 1 ) is best possible.

For s = 0, some results of this paragraph are strengthened in the following one in the sense that with the same hypotheses, we will obtain stronger conclusions.

e ( G ) - (dc(x> + dc(y)) + 1 .


111.4 Property P: G is bipanconnected

The positive integer s being given, and for every integer n 2 3s of the

K I , ~ ) is not bipanconnected and satisfies &;(a) + &(b) 2 n + s for every non-edge ab with a in A and b in B. Therefore we cannot hope to prove any Theorem T 2 ( P , n + s) for a fixed value of s and thus we establish the results T3(P, k ) and T4(P, k ) . However, we will obtain the Theorem T2(P’,n + 2) for the property P’: G is quasi-bipanconnected.

Lemma 3.4.1. Let H be a hamiltonian balanced bipartite graph of order 2p, and let x and y be two extra vertices respectively adjacent to some vertices of H B and H A such that d H ( x ) + d H ( y ) 2 p + 1. Then there exist xy-paths of every odd length between 3 and 2p + 1.

Let C+ be an arbitrarily oriented hamiltonian cycle of H and let {x,}, 1 5 i 5 dH(x), be the neighbors of x on C+. If there exists no xy-path P2/ of length 2t - 1 for some t between 2 and p + 1, let (y,}, 1 5 i 5 dH(x ) , be the vertices of H A respectively lying at the distance 2t - 3 on C+ from the x,’s. The vertex y is adjacent to none of the y I ’s . Thus d H ( y ) 5 p - dH(x) , a contradiction.

Theorem 3.4.2.

same parity as s, the graph: G I O ( ~ ) = K ( n - s ) / 2 , ( n - h ) / 2 + (K(n+s-2)/2,(n+F-2)/2 U

Proof.

I

If 6(G) 2 ( n + 2)/2 then G is bipanconnected.

Let x in A and y in B be two vertices of G. The subgraph H =

G - { x , y } which satisfies 6 ( H ) 2 n/2, is hamiltonian by Theorem 2.1.1. Moreover dH(x) 2 n/2 and d H ( y ) 2 n/2. Thus by Lemma 3.4.1, G contains a xy-path of every odd length between 3 and 2n - 1.

The graphs G I ( l ) if n is odd and G2(1) if n is even show that Theorem T 3 ( P , n + 2) is best possible.

In [3], Bagga and Varma already proved T3(P,n + 3).

Theorem 3.4.3.

Proof.

I

If e (G) 2 (n - 1)’ + n + 1 then G is bipanconnected.

Suppose e(G) 2 (n - I)’ + n + 1 and let x in A and y in B be two vertices of G. By Theorem 3.3.2, there exists a hamiltonian path a l b l a 2 b 2 ~ ~ ~ a n b , in G with x = a l and y = b,7. If G contains no xy- path of odd length L = 2t - 1 for some t between 2 and n - I , then no edge ~ ~ b , + , - ~ for 1 5 i 5 t nor b r a n + l - r + ~ for 1 I i 5 t - 1 can exist. Moreover, for 2 d j 5 n - t , if alb , is an edge of G, then b,ar+J-l is

I)’ + n + 1, a contradiction.

The graph G3(0) of paragraph 11.2 shows that Td(P,n + 1) is best possible. Note that the hypotheses 6(G) 2 ( n + 2)/2 or e (G) 2 (n - 1)’ + n + 1 imply not only that G is bipanconnected but also that the distance d(x,y) between any pair of vertices, x in A and y in B, is at most 3, and that every edge belongs to cycles of every even length.

Proof.

not an edge. Therefore e(G) 5 n’ - t - (t - 1) - (n - t - 1) < ( n - I


If we come back to the graphs Glo(2), we observe that the eventually missing paths of odd length between two vertices are all of length 1 or 3. This leads us to introduce property PI: G is quasi-bipanconnected. The proofs of the previous two theorems show that T3(P ' ,n + 2) and T4(P' ,n + 1) are still true and best possible. For this new property P ' , we obtain the following result of type T2.

Theorem 3.4.4. vertices a in A and b in B, then G is quasi-bipanconnected.

If & ( a ) + dc(b) 2 n + 2 for every pair of non-adjacent

Proof. The hypothesis implies G = Kn,n or n 2 4 and 6 2 3. We use induction on n. The property is true for n 5 3. Suppose it is true until order 2(n - 1) and consider a balanced bipartite graph G of order 2n satisfying the hypothesis. Let x in A and y in B be two vertices of G and let H = G - { x , y } . SincedH(u) + d ~ ( b ) 2 n = n ( H ) + 1 for every non- edge a b of H with a in H f l A and b in H fl B , the graph His hamiltonian. We have the following two cases:

(1) If &(x) + d H ( y ) 2 n = n ( H ) + 1, then by Lemma 3.4.1, G contains xy-paths of every odd length between 3 and 2n - 1.

(2) If dH(x) + d H ( y ) 5 n - 1, which implies in particular that x and y are adjacent, we consider the subsets L, = N ( x ) f l H B and L,. =

N ( y ) fl HA.

If the subgraph induced in G by L, U L , is complete bipartite, each non- edge a b of H with a in A and b in B has at most one endvertex adjacent to x or toy and thus d H ( a ) + d H ( b ) 2 n + 1 = n ( H ) + 2. By the induction hypothesis, H is quasi-bipanconnected. Hence, if z is a vertex of L , and t a vertex of L, , H contains zt-paths of every odd length except perhaps for the length 3. But zt-paths of length 3 also exist in H since L, and L, have each at least two vertices. All these paths together provide in G xy-paths of every odd length between 1 and 2n - 1.

If the subgraph induced by L, U L , is not complete, let zt be a non- edge of G with 7 in L, and t in L, , and F the balanced bipartite graph G-{x, y , z , t } of half order p = n - 2. We will show that the ( p + 1)- biclosure BF,+I of F is complete bipartite. If uu is a non-edge of F with u in FA and u in FB but at most one endvertex u or u in L , U L , , then d F ( u ) + d F ( u ) 2 &(u ) + &(u) - 3 2 n - I = p + 1. So in the construction of B F p t l , we first add all these edges to get F1 (say). Now, if uu is a non-edge of F I with u in L, and u in L,, the degree of u in F , is at least n - d G ( x ) and the degree of' u is at least n - d~ ( y ) . Their degree sum is at least 2n - ( d c ( x ) + d c ( y ) ) 2 2n - (n - 1) 2 p + 1. 'Therefore BF,+l = Kp,p and F is hamiltonian. Since d F ( z ) + dF(t ) 2 n == p + 2, by Lemma 3.4. I , there exist zt-paths in H of every odd length between 3 and 2n - 3. These paths provide in G xy-paths of every odd length between 5 and 2n - 1. The existence of xy-paths of length 3 depends on the existence


of edges inside L, U L,. Anyway the graph G is quasi-bipanconnected and the theorem is true for all n.

The graphs GI(]) if n is odd and Gz(1) if n is even show that the value n + 2 in T 2 ( P ' , n + 2) is best possible.

I

IV. OTHER PROPERTIES.

IV.l Property P: a ~ p ( G ) 5 2s

Theorem 4.1.1. The property P: a B I p ( G ) 5 2s is (2n - 2s - ])-bistable for every integer s with 0 5 s 5 n - 1.

Proof. If for some a in A and b in B, aBlp(G + ab) 5 2s but aYBIP(G) > 2s, then there is a set W of 2s vertices of G such that a , b are not in W and W U {a ,b} is a balanced independent set in G. Hence dc(a) 5 n - 1 - s and & ( b ) 5 n - 1 - s, so dc(a) + dc(b) < 2n- 2s - 1 . 1

The graph G l l ( s ) = Kn-r- l ,n-s- l + Fs+l,5+l shows that T,(P,2n - 2s - 1) are best possible for 0 5 i 5 3. However, for s > 0, T4(P,2n - 2s - 1) is not and the following theorem shows that T4(P,2n - (s + 1)2) is true, and best possible because of G l l ( s ) .

Theorem 4.1.2. If e(G) 2 ( n - 1)' + 2n - ( s + 1)2, then a B I p ( G ) 5 2s.

This is obvious since a ~ , p ( G ) 2 2(s + 1) implies e(G) 5 n2 - Proof. (s + 1)'. 1

IV.2 Property P: G has an s-factor

Theorem 4.2.1. For s 2 2, the property P: G has an s-factor is (n + 2s - 3)-bistable and for s = 1, the property P: G has a 1-factor is n-bistable.

If G + ab has an s-factor, but not G, then G has a bipartite subgraph F = ( A , B , E ( F ) ) with dF(a) = dF(b) = s - 1 and d F ( x ) = s for all vertices x in V(G) - {a , b}. Let M = { y E B : ay E E(G) - E ( F ) } and N = {x E A : xb E E(G) - E ( F ) } . The subgraph F has ns-1 edges, 2(s - 1) of them are incident to a or b. Therefore the number Q of edges of F which are incident with M U N is at most ns-l-2(s - 1) =

(n - 2)s + 1. On the other hand, F contains no edge xy with x E M and y E N for otherwise F - xy + bx + ay would be an s-factor of G. Hence Q = s(lMI + INI) and thus IMI + IN1 5 n - 2 + l/s. If s 2 2 we have IMI + IN1 5 n - 2. Then &(a) + &(b> = d ~ ( a ) +

Proof.


dF(b) + (MI + IN1 < n + 2s - 3 . If s = 1 then & ( a ) -t dc(b) 5 n - l < n . I

- For n 2 3.7, the graph Glz(s) is obtained from (KI,I + 2 K \ - 1 , ~ - 1 ) U K 1 , l U Kn-2s,n-2., by adding some edges: Let us denote by {a ,b} the first by { c , d } the second one, by K\ - I , \ - I and Kf-l , ,- l the two balanced complete bipartite graphs each with 2s - 2 vertices. We add:

-all the edges between c and K y - l , y - l n B and between d and

-one edge a d with a E Kn-2,,n-2, n A . -one edge c p with p E Kn-2s,n-2, f l B. -all the edges between b and Kn-2s,n-25 n A . The graph G12(s) shows, for s 3 2 and n 2 3s, that To(P,n + 2s - 3 )

is best possible for the property P: G has an s-factor. For the property P: G has a I-factor, the graph G13 = K n - ~ , n U K I , ~

shows that T 4 ( P , n ) is best possible. The graphs G14 = Kn/2,(n-2)12 U Kn/2 , (n+2) /2 if n is even and G I ~ = K ( n + 1 ) / 2 , ( n - 1 ) / 2 U K ( ~ - I ) / ~ , ( ~ + I ) / z if n is odd show that T3(P, n ) is best possible. Theorems T , ( P , n ) are thus best possible for 0 5 i 5 4.

K : - ~ , , - , n A.

References

[ l ] D. Amar, A condition for a hamiltonian bipartite graph to be bipancyclic, Discrete Math. 102 (1992) 221 -227.

[2] D. Amar, Bipancyclic Bipartite Hamiltonian graph and number of edges. Rapport de Recherche No. 9 1 - 10. Universitt Bordeaux I. France.

(31 K. S. Bagga and B. N. Varma, Bipartite graphs and degree conditions, Graph Theory, Cornbinatorics, Algorithms, and Applications, (ed. Y. Alavi, F. Chung, R. Graham, D. Hsu), Siam 1991, 564-573.

[4] B. Bollobas, Extremal graph theory, to appear in: Handbook of Cornbi- natorics, Vol. I (ed. R.L. Graham, M. Grotschel and Lovasz), North- Holland, Amsterdam, in press.

[5] J. A. Bondy and V. Chvital, A method in graph theory, Discrete Math. 15 (1976), 11 1 - 136.

[6] J. A. Bondy and U. S. R. Murty, Graph Theory with Applications, Macmillan, London (1976).

171 F. Hasek, The Good Soldier Schweik, Penguin Books, Harmondsworth (1951).

[8] J. Mitchem and E.F. Schmeichel, Bipartite graphs with cycles of all even lengths, J. Graph Theory 6 (1982), 429-439.

[9] J. Moon and L. Moser, On hamiltonian bipartite graphs, Israel J. Math. 1 (1963), 163-165.

B I CLOS U RE AN D B I STAB I LI TY 529

[lo] E. Schmeichel and J. Mitchem, Pancyclic and bipancyclic graphs. A Survey. Graphs and Applications. Proceedings c$ the First Colorado Symposium on Graph Theory (eds. F. Harary and J. S. Maybee), Wiley, New York (1985) 271-278.

Received June 10, 1994

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Biclosure and bistability in a balanced bipartite graph