D:\sbornik\...\Article.DVI


Mathematical Problems of Computer Science 40, 13{22, 2013.

N on-hamiltonian Gr aphs with Given T oughness

Zh o r a G. N iko g h o s ya n 1

Institute for Informatics and Automation Problems of NAS RA

e-mail: zhora@ipia.sci.am

Abstract

In 1973, Chv¶atal introduced the concept of toughness ¿ of a graph and conjec-
tured that there exists a ¯nite constant t0 such that every t0-tough graph (that is
¿ ¸ t0) is hamiltonian. To solve this challenging problem, all e®orts are directed to-
wards constructing non-hamiltonian graphs with toughness as large as possible. The
last result in this direction is due to Bauer, Broersma and Veldman, which states that
for each positive ², there exists a non-hamiltonian graph with 9

4
¡ ² · ¿ < 9

4
. The

following related broad-scale problem, reminding the well-known pancyclicity or hypo-
hamiltonicity, arises naturally: whedher there exists a non-hamiltonian graph with a
given toughness. We conjecture that if there exist a non-hamiltonian t-tough graph
then for each rational number a with 0 < a · t there exists a non-hamiltonian graph
whose toughness is exactly a. In this paper we prove this conjecture for t = 9

4
¡ ² by

using a number of additional modi¯ed building blocks to construct the required graphs.
Keywords: Hamilton cycle, Toughness of graph.

1 . In t r o d u c t io n

On ly ¯ n it e u n d ir e c t e d g r a p h s wit h o u t lo o p s o r m u lt ip le e d g e s a r e c o n s id e r e d . Th e s e t o f
ve r t ic e s o f a g r a p h G is d e n o t e d b y V ( G) a n d t h e s e t o f e d g e s - b y E ( G ) . Th e o r d e r a n d t h e
in d e p e n d e n c e n u m b e r o f G a r e d e n o t e d b y n a n d ®, r e s p e c t ive ly. Fo r S a s u b s e t o f V ( G) ,
we d e n o t e b y GnS t h e s u b g r a p h o f G in d u c e d b y V ( G ) nS. Th e n e ig h b o r h o o d o f a ve r t e x
x 2 V ( G) is d e n o t e d b y N ( x) . A g r a p h G is h a m ilt o n ia n if G c o n t a in s a H a m ilt o n c yc le , i.e .
a c yc le o f le n g t h n. A g o o d r e fe r e n c e fo r a n y u n d e ¯ n e d t e r m s is [5 ].

Th e c o n c e p t o f t o u g h n e s s o f a g r a p h wa s in t r o d u c e d in 1 9 7 3 b y Ch v¶a t a l [6 ]. L e t ! ( G )
d e n o t e t h e n u m b e r o f c o m p o n e n t s o f a g r a p h G. A g r a p h G is t-t o u g h if jSj ¸ t! ( GnS )
fo r e ve r y s u b s e t S o f t h e ve r t e x s e t V ( G) wit h ! ( GnS ) > 1 . Th e t o u g h n e s s o f G, d e n o t e d
¿ ( G) , is t h e m a xim u m va lu e o f t fo r wh ic h G is t-t o u g h ( t a kin g ¿ ( Kn ) = 1 fo r a ll n ¸ 1 ) .
Mu c h o f t h e r e s e a r c h o n t h is s u b je c t h a ve b e e n in s p ir e d b y t h e fo llo win g c o n je c t u r e d u e t o
Ch v¶a t a l [6 ].

Conjectur e 1: There exists a ¯nite constant t0 such that every t0-tough graph is hamil-
tonian.

To s o lve t h is c h a lle n g in g p r o b le m , a ll e ®o r t s a r e d ir e c t e d t o wa r d s c o n s t r u c t in g n o n -
h a m ilt o n ia n g r a p h s wit h t o u g h n e s s a s la r g e a s p o s s ib le . In [6 ], Ch v¶a t a l c o n s t r u c t e d a n
in ¯ n it e fa m ily o f n o n -h a m ilt o n ia n g r a p h s wit h ¿ = 3

2
, a n d t h e n Th o m a s s e n [[4 ]; p: 1 3 2 ] fo u n d

1 G. G. Nicoghossian (up to 1997).

1 3



1 4 Non-hamiltonian Graphs with Given Toughness

n o n -h a m ilt o n ia n g r a p h s wit h ¿ > 3
2
. L a t e r E n o m o t o e t a l. [7 ] h a ve fo u n d n o n -h a m ilt o n ia n

g r a p h s wit h ¿ ¸ 2 ¡ ² fo r e a c h p o s it ive ². Th e la s t r e s u lt in t h is d ir e c t io n is d u e t o B a u e r ,
B r o e r s m a a n d V e ld m a n [2 ] in s p ir e d b y s p e c ia l c o n s t r u c t io n s in t r o d u c e d in [1 ] a n d [3 ].

T heor em A: F or each positive ² > 0 , there exists a non-hamiltonian graph with 9
4
¡ ² <

¿ < 9
4
.

Th e fo llo win g r e la t e d b r o a d -s c a le p r o b le m , r e m in d in g t h e we ll-kn o wn p a n c yc lic it y o r
h yp o h a m ilt o n ic it y, a r is e s n a t u r a lly.

P r oblem: D o e s t h e r e e xis t a n o n -h a m ilt o n ia n g r a p h wit h t h e g ive n t o u g h n e s s ?
Th e fo llo win g m e t a c o n je c t u r e s e e m s r e a s o n a b le .
Conjectur e 2: If there exists a non-hamiltonian t-tough graph then for each rational

number a with 0 < a · t there exists a non-hamiltonian graph whose toughness is exactly
a.

In t h is p a p e r we p r o ve t h is c o n je c t u r e fo r t = 9
4
¡ ².

T heor em 1: F or each rational number t with 0 < t < 9
4
, there exists a non-hamiltonian

graph G with ¿ ( G) = t.
Th e o r e m 1 p r o vid e s a ls o a c o m p le t e b a c kg r o u n d fo r fu r t h e r in ve s t ig a t io n t o wa r d s ¯ n d in g

n o n -h a m ilt o n ia n g r a p h s wit h t o u g h n e s s a t le a s t 9
4
.

2 . P r e lim in a r ie s

To p r o ve Th e o r e m 1 , we n e e d b o t h n e w a n d o ld g r a p h c o n s t r u c t io n s .
De¯nition 1. L e t L(1) b e a g r a p h o b t a in e d fr o m C8 = w1w2:::w8w1 b y a d d in g t h e e d g e s

w2w4; w4w6; w6w8 a n d w2w8. P u t x = w1 a n d y = w5. Th is is t h e we ll-kn o wn b u ild in g b lo c k
L u s e d t o o b t a in ( 9

4
¡ ²) -t o u g h n o n -h a m ilt o n ia n g r a p h s ( s e e Fig u r e 1 in [2 ]) .

In t h is p a p e r we will u s e a n u m b e r o f a d d it io n a l m o d ī e d b u ild in g b lo c ks .
De¯nition 2: L et L(2) be the graph obtained from L(1) by deleting the edges w1w2, w2w8

and identifying w2 with w8.
De¯nition 3: L et L(3) be the graph obtained from L(1) by adding a new vertex w9 and

the edges w4w9, w6w9.
De¯nition 4: L et L(4) be the graph obtained from the triangle w1w2w3w1 by adding the

vertices w4; w5 and the edges w1w4, w3w5. P ut x = w4 and y = w5.
De¯nition 5: F or each L 2 fL(1); L(2)g, de¯ne the graph G( L; x; y; l; m ) ( l; m 2 N ) as

follows. Take m disjoint copies L1; L2; :::; Lm of L, with xi; yi the vertices in Li corresponding
to the vertices x and y in L ( i = 1 ; 2 ; :::; m) . L et Fm be the graph obtained from L1 [ ::: [ Lm
by adding all possible edges between the pairs of vertices in x1; :::; xm; y1; :::; ym. L et T = Kl
and let G ( L; x; y; l; m ) be the join T _ Fm of T and Fm.

Th e fo llo win g c a n b e c h e c ke d e a s ily.
Claim 1: The vertices x and y are not connected by a Hamilton path of L(i) ( i = 1 ; 2 ; 3 ) .
Th e p r o o f o f t h e fo llo win g r e s u lt o c c u r s in [1 ].
Claim 2: L et H be a graph and x; y two vertices of H which are not connected by a

Hamilton path of H. If m ¸ 2 l + 1 then G ( H; x; y; l; m) is non-hamiltonian.

3 . P r o o f o f Th e o r e m 1

B y t h e d e ¯ n it io n , t h e t o u g h n e s s ¿ ( G) is a r a t io n a l n u m b e r . L e t t b e a n y r a t io n a l n u m b e r
wit h 0 < t < 9

4
a n d le t t = a

b
fo r s o m e in t e g e r s a; b.



Zh. Nikoghosyan 1 5

Case 1: 0 < a
b

< 1 .
L e t Ka;b b e t h e c o m p le t e b ip a r t it e g r a p h G = ( V1; V2; E ) wit h ve r t e x c la s s e s V1 a n d V2

s u c h t h a t jV1j = a a n d jV2j = b. S in c e ab < 1 , we h a ve ®( G ) = b > ( a + b ) = 2 a n d t h e r e fo r e ,
Ka;b is a n o n -h a m ilt o n ia n g r a p h . Cle a r ly, ¿ · jV1j=! ( GnV1 ) = a=b. Ch o o s e S ½ V ( G) s u c h
t h a t ¿ ( G) = jSj=! ( GnS ) . P u t S \ Vi = Si a n d jSij = si ( i = 1 ; 2 ) . If VinS 6= ; ( i = 1 ; 2 )
t h e n c le a r ly ! ( GnS ) = 1 , wh ic h is im p o s s ib le b y t h e d e ¯ n it io n . H e n c e , VinS = ; fo r s o m e
i 2 f1 ; 2 g.

Case 1.1: i = 2 .
It fo llo ws t h a t

¿ =
b + s1
a ¡ s1

¸ b
a
:

R e c a llin g t h a t ¿ · a
b
, we h a ve b2 · a2, c o n t r a d ic t in g t h e h yp o t h e s is a

b
< 1 .

Case 1.2: i = 1 .
It fo llo ws t h a t

¿ =
s2 + a

b ¡ s2
¸ a

b
;

im p lyin g im m e d ia t e ly t h a t ¿ = a
b
.

Case 2: a
b

= 1 .
L e t G b e a g r a p h o b t a in e d fr o m C6 = v1v2:::v6v1 b y a d d in g a n e w ve r t e x v7 a n d t h e e d g e s

v1v7; v4v7; v2v6. Cle a r ly, G is n o t h a m ilt o n ia n a n d ¿ ( G) = 1 .
Case 3: 1 < a

b
< 3

2
.

Case 3.1: a
b

< 3
2
¡ 1

b
.

L e t V1; V2; V3 b e p a ir wis e d is jo in t s e t s o f ve r t ic e s :

V1 = fx1; x2; :::; xa¡b+1g; V2 = fy1; y2; :::; ybg; V3 = fz1; z2; :::; zbg:

Jo in e a c h xi t o a ll t h e o t h e r ve r t ic e s a n d e a c h zi t o e ve r y o t h e r zj a s we ll a s t o t h e ve r t e x
yi wit h t h e s a m e s u b s c r ip t i. Ca ll t h e r e s u lt in g g r a p h G. Ch o o s e W ½ V ( G) s u c h t h a t
¿ ( G) = jW j=! ( GnW ) . P u t m = jW \ V3j. Cle a r ly, W is a m in im a l s e t wh o s e r e m o va l fr o m
G r e s u lt s in a g r a p h wit h ! ( GnW ) c o m p o n e n t s . A s W is a c u t s e t , we h a ve V1 ½ W a n d
m ¸ 1 . Fr o m t h e m in im a lit y o f W we e a s ily c o n c lu d e t h a t V2 \ W = ; a n d m · b ¡ 1 . Th e n
we h a ve jW j = m + a ¡ b + 1 a n d ! ( GnW ) = m + 1 . H e n c e ,

¿ ( G ) =
jW j

! ( GnW )
= m in

1·m·b¡1
m + a ¡ b + 1

m + 1
=

a

b
:

To s e e t h a t G is n o n -h a m ilt o n ia n , le t u s a s s u m e t h e c o n t r a r y, i.e . le t C b e a H a m ilt o n c yc le
in G. D e n o t e b y F t h e s e t o f e d g e s o f C h a vin g a t le a s t o n e e n d ve r t e x in V2. S in c e V2 is
in d e p e n d e n t , we h a ve jF j = 2 jV2j. On t h e o t h e r h a n d , t h e r e a r e a t m o s t 2 jV1j e d g e s in F
h a vin g o n e e n d ve r t e x in V1 a n d a t m o s t jV3j e d g e s in F h a vin g o n e e n d ve r t e x in V3. Th u s ,

2 b = 2 jV2j = jF j · 2 jV1j + jV3j = 2 ( a ¡ b + 1 ) + b = 2 a ¡ b + 2 :

B u t t h is is e qu iva le n t t o a=b ¸ 3 =2 ¡ 1 =b, c o n t r a d ic t in g t h e h yp o t h e s is .
Case 3.2: a

b
¸ 3

2
¡ 1

b
.

B y c h o o s in g a s u ± c ie n t ly la r g e q 2 N wit h
a

b
=

aq

bq
<

3

2
¡ 1

bq
;



1 6 Non-hamiltonian Graphs with Given Toughness

we c a n a r g u e a s in Ca s e 3 .1 .
Case 4: a

b
= 3

2
.

A n e xa m p le o f a n o n -h a m ilt o n ia n g r a p h wit h ¿ = 3 =2 is o b t a in e d wh e n in t h e P e t e r s e n
g r a p h , e a c h ve r t e x is r e p la c e d b y a t r ia n g le .

Case 5: 3
2

< a
b

< 7
4
.

Claim 3: F or l ¸ 2 and m ¸ 1 ,

¿
³
G

³
L(2); x; y; l; m

´´
=

l + 3 m

1 + 2 m
:

P r oof. L e t G = G ( L(2); x; y; l; m) fo r s o m e l ¸ 2 a n d m ¸ 1 . Ch o o s e S µ V ( G) s u c h t h a t

! ( GnS ) > 1 ; ¿ ( G ) = jSj
! ( GnS )

:

Ob vio u s ly, V ( T ) µ S. D e ¯ n e Si = S \ V ( Li ) , si = jSij, a n d le t !i b e t h e n u m b e r o f
c o m p o n e n t s o f LinSi t h a t c o n t a in n e it h e r xi n o r yi ( i = 1 ; :::; m ) . Th e n

¿ ( G ) =
l +

Pm
i=1 si

c +
Pm

i=1 !i
¸ l +

Pm
i=1 si

1 +
Pm

i=1 !i
;

wh e r e

c =

(
0 if xi; yi 2 S fo r a ll i 2 f 1 ; :::; mg
1 o t h r wis e .

It is e a s y t o s e e t h a t

!i · 2 ; si ¸
3

2
!i ( i = 1 ; :::; m ) :

Th e n

¿ ¸
l + 3

2

Pm
i=1 !i

1 +
Pm

i=1 !i
=

l ¡ 3
2

1 +
Pm

i=1 !i
+

3

2

¸
l ¡ 3

2

1 + 2 m
+

3

2
=

l + 3 m

1 + 2 m
:

S e t U = V ( T ) [ U1 [ ::: [ Um, wh e r e Ui is t h e s e t o f ve r t ic e s o f Li wit h t h e d e g r e e a t le a s t
4 in Li ( i = 1 ; :::; m ) . Th e p r o o f o f Cla im 3 is c o m p le t e d b y o b s e r vin g t h a t

¿ ( G) · jUj
! ( GnU ) =

l + 3 m

2 m + 1
:

Case 5.1:b = 2 k + 1 for some integer k.
Co n s id e r t h e g r a p h G( L(2); x; y; a ¡ 3

2
( b ¡ 1 ) ; b¡1

2
) .

Case 5.1.1: a
b
· 7

4
¡ 9

4b
.

B y t h e h yp o t h e s is ,

m =
b ¡ 1

2
¸ 2

µ
a ¡ 3

2
( b ¡ 1 )

¶
+ 1 = 2 l + 1 :

B y Cla im 2 , G is n o t h a m ilt o n ia n . Cle a r ly b ¸ 3 , im p lyin g t h a t m = ( b ¡ 1 ) =2 ¸ 1 . If
a
b
¸ 3

2
+ 1

2b
t h e n l = a ¡ 3

2
( b ¡ 1 ) ¸ 2 a n d b y Cla im 3 , ¿ ( G ) = a

b
. N o w le t a

b
< 3

2
+ 1

2b
. B y

c h o o s in g a s u ± c ie n t ly la r g e in t e g e r q wit h

a

b
=

aq

bq
¸ 3

2
+

1

2 bq
;



Zh. Nikoghosyan 1 7

we c a n a r g u e a s in t h e p r e vio u s c a s e .
Case 5.1.2: a

b
> 7

4
¡ 9

4b
.

B y c h o o s in g a s u ± c ie n t ly la r g e in t e g e r q wit h

a

b
=

aq

bq
· 7

4
¡ 9

4 bq
;

we c a n a r g u e a s in Ca s e 5 .1 .1 .
Case 5.2: b = 2 k for some integer k.
Co n s id e r t h e g r a p h G0 o b t a in e d fr o m G( L(2); x; y; l; m ) b y r e p la c in g Lm wit h L

(3).
Claim 4: F or l ¸ 2 and m ¸ 1 ,

¿ ( G0 ) =
l + 3 m + 1

2 ( m + 1 )
:

P r oof: Ch o o s e S µ V ( G0 ) s u c h t h a t ! ( G0nS ) > 1 a n d ¿ ( G0 ) = jSj=! ( G0nS ) . Ob vio u s ly,
V ( T ) µ S. D e ¯ n e Si = S \ V ( Li ) , si = jSij, a n d le t !i b e t h e n u m b e r o f c o m p o n e n t s o f
LinSi t h a t c o n t a in n e it h e r xi n o r yi ( i = 1 ; :::; m ) . S in c e si ¸ 32 !i ( i = 1 ; :::; m ¡ 1 ) a n d
sm ¸ 43 !m, we h a ve

¿ ( G0 ) ¸ l +
Pm

i=1 si
c +

Pm
i=1 !i

¸
l + 3

2

Pm¡1
i=1 !i +

4
3
!m

1 +
Pm

i=1 !i
=

l ¡ 1
6
!m

1 +
Pm

i=1 !i
+

3

2
;

wh e r e c = 0 if xi; yi 2 S fo r a ll i 2 f 1 ; :::; mg a n d c = 1 , o t h e r wis e . Ob s e r vin g a ls o t h a t
!i · 2 ( i = 1 ; :::; m ¡ 1 ) a n d !m · 3 , we o b t a in

( l ¡ 2 )
mX

i=1

!i +
1

3
( m + 1 ) !m · ( l ¡ 2 ) ( 2 m + 1 ) + ( m + 1 ) · 2 l ( m + 1 ) :

B u t t h is is e qu iva le n t t o
l ¡ 1

6
!m

1 +
Pm

i=1 !i
+

3

2
¸ l ¡ 2

2 ( m + 1 )
+

3

2
;

im p lyin g t h a t

¿ ( G0 ) ¸ l ¡ 2
2 ( m + 1 )

+
3

2
=

l + 3 m + 1

2 ( m + 1 )
:

S e t U = V ( T ) [ U1 [ ::: [ Um, wh e r e Ui is t h e s e t o f ve r t ic e s o f Li wit h t h e d e g r e e a t le a s t
4 in Li ( i = 1 ; :::; m ) . Th e p r o o f o f Cla im 4 is c o m p le t e d b y o b s e r vin g t h a t

¿ ( G0 ) · jUj
! ( GnU ) =

l + 3 m + 1

2 ( m + 1 )
:

Co n s id e r t h e g r a p h G0 wit h m = b
2
¡ 1 a n d l = a ¡ 3

2
b + 2 . Cle a r ly m = b

2
¡ 1 ¸ 1 a n d

l = a ¡ 3
2
b + 2 ¸ 2 . B y Cla im 4 , ¿ ( G0 ) = a

b
. If a

b
· 7

4
¡ 3

b
t h e n m ¸ 2 l + 1 , a n d b y Cla im 2 ,

G0 is n o t h a m ilt o n ia n . Ot h e r wis e , b y c h o o s in g a s u ± c ie n t ly la r g e q wit h

a

b
=

aq

bq
· 7

4
¡ 3

b
;

we c a n a r g u e a s in t h e p r e vio u s c a s e .
Case 6: 7

4
¡ ² < a

b
· 2 .



1 8 Non-hamiltonian Graphs with Given Toughness

L e t m = m1 + m2 ¸ 2 l + 1 a n d le t G00 b e t h e g r a p h o b t a in e d fr o m G ( L(1); x; y; l; m ) b y
r e p la c in g Li wit h L

(2) ( i = m1 + 1 ; m1 + 2 ; :::; m ) . B y Cla im 2 , G
00 is n o t h a m ilt o n ia n .

Claim 5: F or l ¸ 2 , m ¸ 1 and m2 ¸ l ¡ 2 ,

¿ ( G00 ) =
l + 3 m2
2 m2 + 1

:

P r oof: Ch o o s e S µ V ( G00 ) s u c h t h a t ¿ ( G00 ) = jSj=! ( G00nS ) . Ob vio u s ly, V ( T ) µ S.
D e ¯ n e Si = S\V ( Li ) , si = jSij, a n d le t !i b e t h e n u m b e r o f c o m p o n e n t s o f LinSi t h a t c o n t a in
n e it h e r xi n o r yi ( i = 1 ; :::; m ) . S in c e si ¸ 2 !i ( i = 1 ; :::; m1 ) a n d si ¸ 32 !i ( i = m1 + 1 ; :::; m) ,
we h a ve

¿ ( G00 ) ¸
l +

Pm1
i=1 si +

Pm
i=m1+1

si
c +

Pm
i=1 !i

¸
l + 2

Pm1
i=1 !i +

3
2

Pm
i=m1+1

!i
1 +

Pm
i=1 !i

l + 1
2

Pm1
i=1 !i ¡ 32 +

3
2
( 1 +

Pm
i=1 !i )

1 +
Pm

i=1 !i
=

2 l +
Pm1

i=1 !i ¡ 3
2 ( 1 +

Pm
i=1 !i )

+
3

2
;

wh e r e c = 0 if xi; yi 2 S fo r a ll i 2 f 1 ; :::; mg a n d c = 1 , o t h e r wis e . Ob s e r vin g t h a t !i · 2
( i = 1 ; :::; m ) , we o b t a in

( 2 l ¡ 3 )
mX

i=m1+1

!i ¡ ( 2 m2 ¡ 2 l + 4 )
m1X

i=1

!i · 4 lm2 ¡ 6 m2:

B u t t h is is e qu iva le n t t o

2 l +
Pm1

i=1 !i ¡ 3
2 ( 1 +

Pm
i=1 !i )

+
3

2
¸ 2 l ¡ 3

2 ( 2 m2 + 1 )
+

3

2
;

im p lyin g t h a t

¿ ( G00 ) ¸ 2 l ¡ 3
2 ( 2 m2 + 1 )

+
3

2
=

l + 3 m2
2 m2 + 1

:

S e t U = V ( T ) [ U1 [ ::: [ Um, wh e r e Ui is t h e s e t o f ve r t ic e s o f Li wit h t h e d e g r e e a t le a s t 4
in Li ( i = 1 ; :::; m ) . Th e p r o o f o f Cla im 5 is c o m p le t e d b y o b s e r vin g t h a t

¿ ( G00 ) · jUj
! ( GnU ) =

l + 3 m2
2 m2 + 1

:

Case 6.1: b = 2 k + 1 for some integer k.
Co n s id e r t h e g r a p h G00 wit h m2 =

b¡1
2

a n d l = a ¡ 3
2
( b ¡ 1 ) .

Case 6.1.1: a
b
¸ 3

2
+ 1

2b
.

S in c e a
b
· 2 , we h a ve

m2 =
b ¡ 1

2
¸ a ¡ 3

2
( b ¡ 1 ) ¡ 2 = l ¡ 2 :

N e xt , s in c e a
b
¸ 3

2
+ 1

2b
, we h a ve l = a ¡ 3

2
( b ¡ 1 ) ¸ 2 . B y Cla im 5 , ¿ ( G00 ) = a

b
.

Case 6.1.2: a
b

< 3
2

+ 1
2b

.
B y c h o o s in g a s u ± c ie n t ly la r g e in t e g e r q wit h

a

b
=

aq

bq
¸ 3

2
+

1

2 bq
;



Zh. Nikoghosyan 1 9

we c a n a r g u e a s in Ca s e 6 .1 .1 .
Case 6.2: b = 2 k for some integer k.
Co n s id e r t h e g r a p h G000 o b t a in e d fr o m G00 b y r e p la c in g Lm wit h L

(3).
Claim 6: Fo r l ¸ 2 , m ¸ 1 a n d m2 ¸ l ¡ 2 ,

¿ ( G000 ) =
l + 3 m2 + 1

2 ( m2 + 1 )
:

P r oof: Ch o o s e S µ V ( G000 ) s u c h t h a t ¿ ( G000 ) = jSj=! ( G000nS ) . Ob vio u s ly, V ( T ) µ S.
D e ¯ n e Si = S\V ( Li ) , si = jSij, a n d le t !i b e t h e n u m b e r o f c o m p o n e n t s o f LinSi t h a t c o n t a in
n e it h e r xi n o r yi ( i = 1 ; :::; m) . S in c e si ¸ 2 !i ( i = 1 ; :::; m1 ) , si ¸ 32 !i ( i = m1 + 1 ; :::; m ¡ 1 )
a n d sm ¸ 43 !m, we h a ve

¿ ( G000 ) ¸
l +

Pm1
i=1 si +

Pm¡1
i=m1+1

si + sm
c +

Pm
i=1 !i

¸
l + 2

Pm1
i=1 !i +

3
2

Pm¡1
i=m1+1

!i +
4
3
!m

1 +
Pm

i=1 !i

=
l + 1

2

Pm1
i=1 !i ¡ 16 !m + (

3
2

Pm1
i=1 !i +

3
2

Pm
i=m1+1

)

1 +
Pm

i=1 !i

=
l + 1

2

Pm1
i=1 !i ¡ 16 !m

1 +
Pm

i=1 !i
+

3

2
;

wh e r e c = 0 if xi; yi 2 S fo r a ll i 2 f 1 ; :::; mg a n d c = 1 , o t h e r wis e . Ob s e r vin g t h a t !i · 2
( i = 1 ; :::; m ¡ 1 ) a n d !m · 3 , we o b t a in

( l ¡ 2 )
mX

i=m1+1

!i +
1

3
( m2 + 1 ) !m ¡ ( m2 ¡ l + 3 )

m1X

i=1

!i · l + 2 lm2 + 2 :

B u t t h is is e qu iva le n t t o

l + 1
2

Pm1
i=1 !i ¡ 16 !m

1 +
Pm

i=1 !i
+

3

2
¸ l ¡ 2

2 ( m2 + 1 )
+

3

2
;

im p lyin g t h a t

¿ ( G000 ) ¸ l ¡ 2
2 ( m2 + 1 )

+
3

2
=

l + 3 m2 + 1

2 ( m2 + 1 )
:

S e t U = V ( T ) [ U1 [ ::: [ Um, wh e r e Ui is t h e s e t o f ve r t ic e s o f Li wit h t h e d e g r e e a t le a s t 4
in Li ( i = 1 ; :::; m ) . Th e p r o o f o f Cla im 6 is c o m p le t e d b y o b s e r vin g t h a t

¿ ( G000 ) · jUj
! ( GnU ) =

l + 3 m2 + 1

2 ( m2 + 1 )
:

Co n s id e r t h e g r a p h G000 wit h m2 =
b
2
¡ 1 a n d l = a ¡ 3

2
b + 2 .

Case 6.2.1: a
b
· 2 ¡ 1

b
.

B y t h e h yp o t h e s is , m2 =
b
2
¡ 1 ¸

³
a ¡ 3

2
b + 2

´
¡ 2 = l ¡ 2 . N e xt , s in c e a

b
> 7

4
¡ ² > 3

2
,

we h a ve l = 3
2
b + 2 ¸ 2 . B y Cla im 6 , ¿ ( G000 ) = a

b
.



2 0 Non-hamiltonian Graphs with Given Toughness

Case 6.2.2: a
b

> 2 ¡ 1
b
.

B y c h o o s in g a s u ± c ie n t ly la r g e in t e g e r q wit h a
b

= aq
bq

· 2 ¡ 1
bq

, we c a n a r g u e a s in Ca s e
6 .2 .1 .

Case 7: 2 < a
b

< 9
4
.

Case 7.1: b = 2 k + 1 for some integer k.
Case 7.1.1: a

b
· 9

4
¡ 11

4b
.

Ta ke t h e g r a p h G
³
L(1); x; y; a ¡ 2 b + 2 ; b¡1

2

´
. S in c e a

b
> 2 , we h a ve l = a ¡ 2 b + 2 ¸ 2 .

N e xt , t h e h yp o t h e s is a
b
· 9

4
¡ 11

4b
is e qu iva le n t t o

m =
b ¡ 1

2
¸ 2 ( a ¡ 2 b + 2 ) + 1 = 2 l + 1 :

B y Cla im 1 , G
³
L(1); x; y; a ¡ 2 b + 2 ; b¡1

2

´
is n o t h a m ilt o n ia n . Th e t o u g h n e s s

¿
³
G

³
L(1); x; y; a ¡ 2 b + 2 ; b¡1

2

´´
c a n b e d e t e r m in e d e xa c t ly a s in t h e p r o o f o f Th e o r e m A

[2 ],

¿

Ã
G

Ã
L(1); x; y; a ¡ 2 b + 2 ; b ¡ 1

2

!!
¸ l + 4 m

2 m + 1
=

a

b
:

Case 7.1.2: a
b

> 9
4
¡ 11

4b
.

B y c h o o s in g a s u ± c ie n t ly la r g e in t e g e r q wit h

aq

bq
=

a

b
· 9

4
¡ 1 1

4 bq
;

we c a n a r g u e a s in Ca s e 7 .1 .1 .
Case 7.2: b = 2 k for some positive integer k.
Ta ke t h e g r a p h G0000 o b t a in e d fr o m G

³
L(1); x; y; a ¡ 2 b + 2 ; b

2

´
b y r e p la c in g Lm wit h L

(4).

S in c e a
b

> 2 , we h a ve l = a ¡ 2 b + 2 > 2 . W e h a ve a ls o m = b
2

> 1 , s in c e b ¸ 3 .
Claim 7: F or l ¸ 2 and m ¸ 1 ,

¿ ( G0000 ) =
l + 4 m ¡ 2

2 m
:

P r oof: Ch o o s e S µ V ( G000 ) s u c h t h a t ¿ ( G000 ) = jSj=! ( G000nS ) . Ob vio u s ly, V ( T ) µ S.
D e ¯ n e Si = S \ V ( Li ) , si = jSij, a n d le t !i b e t h e n u m b e r o f c o m p o n e n t s o f LinSi t h a t
c o n t a in n e it h e r xi n o r yi ( i = 1 ; :::; m) . S in c e si ¸ 2 !i ( i = 1 ; :::; m) , !i · 2 ( i = 1 ; :::; m ¡ 1 )
a n d !m · 1 , we h a ve

¿ ( G0000 ) =
l +

Pm
i=1 si

c +
Pm

i=1 !i
¸ l + 2

Pm
i=1 !i

1 +
Pm

i=1 !i

=
l ¡ 2

1 +
Pm

i=1 !i
+ 2 ¸ l ¡ 2

2 m
+ 2 =

l + 4 m ¡ 2
2 m

;

wh e r e c = 0 if xi; yi 2 S fo r a ll i 2 f 1 ; :::; mg a n d c = 1 , o t h e r wis e . S e t U = V ( T ) [ U1 [
::: [ Um, wh e r e Ui is t h e s e t o f ve r t ic e s o f Li wit h t h e d e g r e e a t le a s t 4 in Li ( i = 1 ; :::; m) .
Th e p r o o f o f Cla im 7 is c o m p le t e d b y o b s e r vin g t h a t

¿ ( G0000 ) · jUj
! ( GnU ) =

l + 4 m ¡ 2
2 m

:

Case 7.2.1: a
b
· 9

4
¡ 3

b
.



Zh. Nikoghosyan 2 1

B y t h e h yp o t h e s is ,

m ¡ 1 = b
2
¡ 1 ¸ 2 ( a ¡ 2 b + 2 ) + 1 = 2 l + 1 :

B y Cla im 2 , G0000 is n o t h a m ilt o n ia n a n d b y Cla im 7 , ¿ ( G0000 ) = a
b
.

Case 7.2.2: a
b

> 9
4
¡ 3

b
.

B y c h o o s in g a s u ± c ie n t ly la r g e in t e g e r q wit h

aq

bq
=

a

b
· 9

4
¡ 3

3 bq
;

we c a n a r g u e a s in Ca s e 7 .2 .1 . Th e o r e m 1 is p r o ve d .

Refer ences

[1 ] D . B a u e r , H .J. B r o e r s m a , J. va n d e n H e u ve l a n d H .J. V e ld m a n , \ On h a m ilt o n ia n p r o p -
e r t ie s o f 2 -t o u g h g r a p h s " , J . Graph Theory, vo l. 1 8 , p p . 5 3 9 -5 4 3 , 1 9 9 4 .

[2 ] D . B a u e r , H .J. B r o e r s m a a n d H .J. V e ld m a n , \ N o t e ve r y 2 -t o u g h g r a p h is h a m ilt o n ia n " ,
D iscrete Appl. M ath., vo l. 9 9 , p p . 3 1 7 { 3 2 1 , 2 0 0 0 .

[3 ] D . B a u e r a n d E . S c h m e ic h e l, \ To u g h n e s s , m in im u m d e g r e e a n d t h e e xis t e n c e o f 2 -
fa c t o r s " , J . Graph Theory, vo l. 1 8 , p p . 2 4 1 -2 5 6 , 1 9 9 4 .

[4 ] J. C. B e r m o n d , Selected Topics in Graph Theory, in : L . B e in e ke , R .J. W ils o n ( e d s ) ,
A c a d e m ic P r e s s , L o n d o n a n d N e w Y o r k, 1 9 7 8 .

[5 ] J.A . B o n d y a n d U .S .R . Mu r t y, Graph Theory with Applications, Ma c m illa n , L o n d o n
a n d E ls e vie r , N e w Y o r k, 1 9 7 6 .

[6 ] V . Ch v¶a t a l, \ To u g h g r a p h s a n d h a m ilt o n ia n c ir c u it s " , D iscrete M ath., vo l. 5 , p p . 2 1 5 -
2 2 8 , 1 9 7 3 .

[7 ] H . E n o m o t o , B . Ja c ks o n , P . K a t e r in is a n d A . S a it o , \ To u g h n e s s a n d t h e e xis t e n c e o f
k-fa c t o r s " , J . Graph Theory, vo l. 9 , p p . 8 7 -9 5 , 1 9 8 5 .

Submitted 05.09.2013, accepted 18.10.2013.

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Ä. ÜÇÏáÕáëÛ³Ý

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2 2 Non-hamiltonian Graphs with Given Toughness

ËݹÇñÝ»ñÁ. ·áÛáõÃÛáõÝ áõÝÇ, ³ñ¹Ûáù áã ѳÙÇÉïáÝÛ³Ý ·ñ³ý ïñí³Í ÏáßïáõÃÛ³Ùµ: Àëï
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å³ïϳÝáÕ Ï³Ù³Û³Ï³Ý a é³óÇáÝ³É ÃíÇ Ñ³Ù³ñ ·áÛáõÃÛáõÝ áõÝÇ áã ѳÙÇÉïáÝÛ³Ý ·ñ³ý,
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Æ. Íèêîãîñÿí

Àííîòàöèÿ

Ïîíÿòèå æåñòêîñòè ¿ ( G ) ãðàôà G áûëî ââåäåíî Õâàòàëîì â 1973 ãîäó.
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êàæäûé t0-æåñòêèé ãðàô (ýòî îçíà÷àåò, ÷òî ¿ ¸ t0) ãàìèëüòîíîâ. Äëÿ
ðåøåíèÿ ýòîé ñòèìóëèðóþùåé ïðîáëåìû âñå óñèëèÿ êîíöåíòðèðîâàëèñü íà
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âíèìàíèÿ íà ãðàôû ñ íèçêîé æåñòêîñòüþ. Ïîñëåäíèé ðåçóëüòàò â ýòîì
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¡ " · ¿ < 9

4
. Ïîäîáíî çàäà÷àì ïàíöèêëè÷íîñòè è ñóùåñòâîâàíèÿ

ãèïîãàìèëüòîíîâûõ ãðàôîâ, ìû ðàññìàòðèâàåì áîëåå åìêóþ çàäà÷ó: ñóùåñòâóåò
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ñóùåñòâóåò íåãàìèëüòîíîâ t-æåñòêèé ãðàô, òî äëÿ ëþáîãî ðàöèîíàëüíîãî ÷èñëà
a èç èíòåðâàëà ( 0 ; t], ñóùåñòâóåò íåãàìèëüòîíîâ ãðàô, æåñòêîñòü êîòîðîãî ðàâíà
a. Â äàííîé ðàáîòå ìû äîêàçûâàåì ýòó ãèïîòåçó äëÿ ëþáîãî ðàöèîíàëüíîãî
÷èñëà èç èíòåðâàëà ( 0 ; 9

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íåêîòîðûå íîâûå äîïîëíèòåëüíûå êîíñòðóêòèâíûå áëîêè.