A Note on Isomorphic Generalized Prisms

Annals of Discrete Mathematics 27 (1985) 209-214 0 Elsevier Science Publishers B.V. (North-Holland)

209

A NOTE ON ISOMORPHIC GENERALIZED PRISMS Mohanty and D a l j i t Rao

S.P.

Department of Mathematics I . I . T . Kanpur-208016 U.P., I N D I A INTRODUCTION AND P R E L I M I N A R I E S

Throughout

I n a d d i t i o n , we w i l l c o n s i d e r o n l y t h o s e g r a p h s which a r e

m u l t i p l e edges. connected.

w i l l d e n o t e a f i n i t e u n d i r e c t e d graph w i t h o u t l o o p s o r

G

Most graph t h e o r e t i c terms can be found i n Behzad and Chartrand [ 2 ]

o r F. Harary [ 5 ] .

The c e n t r a l c o n c e p t s and n o t a t i o n s of t h i s paper a r e now

defined. If graph

i s a p e r m u t a t i o n of t h e s e t of v e r t i c e s

TI

of

,

i s o b t a i n e d by t a k i n g two d i s j o i n t c o p i e s ,

(G,n)

l a b e l l e d graph n(v$

V(G)

G ,

then the permutation and

G1

t o g e t h e r w i t h edges j o i n i n g t h e v e r t e x

v.

The concept of p e r m u t a t i o n graph of a graph

G2.

G2,

of G

G1

of t h e to

was i n t r o d u c e d

i n 1967 by Chartrand and Harary i n 1 4 1 . I n t h e d e f i n i t i o n of p e r m u t a t i o n graph we w i l l use t h e f o l l o w i n g n o t a t i o n

i n case in

(C,,IT)

and

1 , 2 , ...,n

The l a b e l l i n g

G = (C,,T).

l a b e l l i n g of

along the cycle

V(C )

w i l l be denoted by

Ca

and

of

G

w i l l be c o n s e c u t i v e

The two c o p i e s

Cn. Cb

G1

and

G2

r e s p e c t i v e l y so t h a t

ala2.*.anal b a r e c o n s e c u t i v e l a b e l l i n g s a l o n g t h e c y c l e s Ca and 'b * n l and Cb w i l l be c a l l e d t h e a-cycle and b-cycle of G r e s p e c t i v e l y .

Ca

Klee [ 8 ] c a l l s

(C

a g e n e r a l i z e d n-prism.

,T)

By r(G) we d e n o t e t h e automorphism group of G a c t i n g on V ( G ) = { 1 , 2 , I t i s w e l l known t h a t T(C ) i s D

n'

m e t r i c group on n symbols a s u s u a l . f

'n

blb 2...b

Also

n2

of

Sn a s

IT

t h e d i h e d r a l group.

Let S

...,n}.

d e n o t e t h e sym-

We t a k e composition of two p e r m u t a t i o n s IT

n ( i ) = IT (T (i)). 1 2 1 2

1'

I n [ 6 ] Hedetniemi proved t h a t ( G , T r ' ) i s isomorphic t o (G,n) f o r a l l IT' i n 1 Holton and S t a c e y [ 71 T ( G ) I T T ( G ) U r(G)n- r(G) and p o s s i b l y f o r o t h e r n' too. proved t h e converse of t h i s r e s u l t f o r Roman Numerals, t h a t i s , t h e g r a p h s (Pn,T),

TI

f

A , where A i s d e f i n e d a s A

A II { i + l : i F A } I t B

=

{ ,2,.

..

=

h:

n

=

n

i EA

( i i+l)

fl

j EB

( j ) } where

, n ) = V(Pn) and t h e t h r e e s u b s e t s of V(P ) a r e

mutually d i s j o i n t . Here w e s h a l l prove t h e converse of H e d e t n i e m i ' s r e s u l t s f o r g e n e r a l i z e d n-prisms

(C,,I),

where

IT

C

A.

F i r s t w e develop a method which d e t e r m i n e s whether

S.P. Mohanty and D.Rao

210

a given permutation Let Define D(a) di

=

=

for i

=

1,2

1

2

r(1)

F(2)

(

TI =

(d ,d ,d3,...,dn)

-

TI(i+l)

=

belongs to

TI'

...

r (Cn)71-%

(C ) U

n(n)

I*

if T(i+l) > r ( i ) or

TI(i)

< ~(i)

if r(if1)

,...,n with i+l taken to be 1 when i

We note that D(n1)

=

D(n2)

= n.

if and only if

T2 =

(i) = n (i) + k for i 5 i 5 n where k E {0,1,2 2 1 is reduced modulo n . A l s o D ( n ) has the property

TI

j' C di & O(mod n) i=j and

and

nl(i)

+k

nT(Cn)

if

.

O(mod n)

s

i=l

nl + k , i.e.,

,...,n-11

1 5 j < j ' 5 n-1

if

n

Z d.

(Cn).

of IT as follows:

- v(i) + n

n(i+l)

r (Cn)TIr

Now we have the following:

,...,dn .

THEOREM 1. Let D(T')

D(n) = (dl,d2,d3

Then

n'

E r(Cn)

is one o f t h e following: (i)

(ii)

(iii) (iv)

,...,d ) , (dk,dk+l ,...,dn,dl,...,dk-l) (dl,d2

for

k

E

{2,3

,...,n},

..

(dn,dn-l,. ,dl) ,

,...,n-d ) .

(n-dl,n-d2

Proof.

(i)

If D ( n ' ) =

71'

where

(ii)

e

=

D(n)

1 n + k = (k+l

where

k

71"

= (n(k)

.

.

+

...

...

Sn.

... ...

2 n(k+l)

{2,3,. . , n 1

E

= TI

2 k+2

is the identity of 1

Take

then n '

...,n-1). n-k+l n-k+2 ... n 1 2 ... k ) TI e TI'E r(cn) r(cn) . n-k+2 ... " )

k where n-k n

so

k

E

{0,1,2,

71

n-k+l ~ ( n ) n(1)

...

~(k-1)

Then

D(n") = (dk,dk+l, ...,dn,dl,d2,...,dk-l) = D(n')

and

TI"

=

1

en (k

2 k+l

.. . ...

n-k+l n

n-k+2 1

... ...

) E

k-1

r(cn)

71

r(cn)

Now

Isomorphic Generalized Prisms where

and

n"(i)

=

n-II(n-i+2)

D(n")

=

(dn,dn-l

Then

,...,d 1) ...

2

1 (n-1

=

TI"

.

i

...

n-2

n-i

D(n')

=

...

Take

D(TI")

and

1 (n-n(l)

TI" =

=

... ...

2 n-.ir(2)

... ...

"n )

2

.*.

i

...

n-i

...

i

...

n-n(i) = D(IT')

n-2

i n-if2

...

...

") 2

* * .

r(cn)

(n-dl,n-d2,. . . ,n-d )

1 (n-1

=

II"

r(cn)

...

2 n

n ) *(l n 1

n-1 1

...

E

(iv)

211

n n-n(n) )

e Er(cn)

TI

.

nr(cn)

Then

.

This completes the proof. In the above theorem (ii) and (iii) are called the cyclic and reverse D(n)

cyclic variations of

.

D(n)

of

respectively and (iv) is called the complement

We have E r(cn) 7 r(cn) u r(Cn)n-' r(c ) ni s a c y c l i c o r a reverse c y c l i c v a r i a t i o n of D ( T ) , D(T

COROLLARY 2. D(II')

if and onzy if

')or t h e i r

complements. Let

C

and

C'

be two vertex disjoint induced n-cycles in G

such that each vertex of vertices of

C

and

i

respectively where

.

If the ci, di

is adjacent to exactly one vertex in

1,2,.,.,n,

=

(C,,TI)

are labelled consecutively along the cycles by

C

C'

=

then the permutation

C'

TI' induced by this

labelling is defined in the natural way, as follows: n'(i)

Let

THEOREM 3 .

nf

E

r(cn) Proof.

Let

has order two. (Cn,n')

when

G =

E

where

c.d =

i k

Then

(Cn,n)

(C , n )

where

II E A .

In either case TI'

k

A .

TI E

.

nr(cn)

=

l'(Cn)

TT

-1 T=TI

T(Cn)

so

E(cn,n)

.

(C IT') if and only if n'

Then

and

.

E

n = e , the identify o r

by Hedetniemi's result

It is easy to see that the theorem holds for n = 3 or copies of

(Cn,n')

a : (Cn,n)

* (C

be called

,TI')

Cc

and

Cd

be an isomorphism.

respectively.

a

u

(Cc)

=

Al U B1 , a

-1

(C ) d

Moreover, each vertex of

=

A2 IJ B2

Al U Bl

,TI) =

Let the two

Let

Let

a-1 (Cc) f' C a = A 1 and u -1 (Cc) n C b = B 1 , C - A = A 2 and -1

4 .

(C

TI

1

and < A . U Bi>

=

Cn

C - B = B2 . for

Then

i=1,2.

is adjacent to exactly one vertex of

A2 U B

2'

S.P. Mohanty and D.Rao

212 Let

a. ai+l... ai+k

the cases B1, bl

E

B1.

A2

If

and

aifl

be a maximal segment of

B2 are analogous).

bi+2

to

and

bi+l

1 1

bi+l

k> 1

Let

If

k=l.

But then

ai+lbi+2

,

E(G)

E

a

So

.

1 1

has two neighbors C4.

=

and

bi+2

B1

E

E

a

i+l

and hence

.

B1*

bi+l

ai+l and bi+l Therefore ,

in

Then 4 I J B > = C4 and therefore, 1 1 E(G) s o that bi+l E B l . Then b

in A1 U B1.

bi+l

Then

is impossible.

b L E(G) i+l i+2

a b E E(G) and hence bi+l E B1 i+l i+l n = 4 . Let a.b. f E ( G ) and aibi+l

.

B2 but with

has two neighbors on this

which is in A 2 , has two neighbors

A1 U B 1 , which is impossible.


then

E(G)

E

is in

ai+2, then A1 U B1 must

Now bi+2

5-cycle which is a contradiction. So

(the proofs for

On the other hand, if

is adjacent to

be the 5-cycle ai,ai+l,ai+2,bi+l,bi.

in Al

First let a.b.

is adjacent to bi+l, then

two neighbors in A1 U B1 which is impossible. is adjacent

Ca

b.

So

E

B1

i

and

Hence n=4.

Finally, let

aibi-l

E

E(G)

so that

bi-l

E

B1.

Then

bi

B2

E

for

otherwise a has two neighbors a and bi in A1 U B 1 . If bi-2 E B i-1 i 2' then bi-l has two neighbors in A2 U B2 which is impossible. Therefore bi-2

E

B1

and

a ,b. i-1 1

E

Thus we see that if

A2 U B

2'

n > 4 , then the n-cycles

4. U B . > , 1

1

i=1,2 continue

in the following way:

so

bl,b2...bi-2bi-lai...ai+kbi+k+l"*bn


al,a2...ai-lbi



that relabelling ai,bi in

we obtain

TI

A1 U B1

as

c

i

B1>

and

a b.

i'

1

E

A2 U B

t

r(cn)

nir(cn)

ii' E

r(Cn)

T

r(cn)

and

u

.

A2 U B

r(cn)(nl)-l

= ff

-1

(C,)

r(cn)

.

, this permutation is so

But this result is not true for general

ir"

and hence

= 71

(C,,T)

.

as

di

But since

as the induced permutation from this labelling also.

A1 U Bl = a-'(Cc) ir"

...bi+kai+k+l...an

T"

where

1

For instance,

(CI0,(3,6,4,2,5,7,9,1,8,10)= (ClO,(3,9,7,5,8,6,1,4,2,10)) but (3,6,4,2,5,7,9,1,8,10) k? r(C,,)(3,9,7,5,8,6,1,4,2,10) r(C10)(7,9,1,8,4,6,3,5,2,10)

r(ClO)

.

r(ClO)

or

However, no such example where

71

is a

product of disjoint transpositions could be constructed. We conclude this note with the following. Our interest in generalized n-prisms is due to the following problem mentioned in [ E l . PROBLEM. Which generalized n-prisms admit a HamiZtonian circuit?

213

Isomorphic Generalized Prisms If

D(n)

contains 1 or

n-1

then obviously

the problem is to find those permutations n but

(Cn,?r) is Hamiltonian. (D(n)

for which

either 1 or

n-1

(Cn,IT) is Hamiltonian.

about D(7)

it was possible to enumerate all such D ( n ) ' s

(that is, corresponding

(Cn,n)'s

(2,2,2,2,2)

if

(3,2,2,3,4,4)

does not contain

Using our earlier discussion

are nonisomorphic)

There are none for n = 3 or 4 and for n > 5

So

which are distinct 3 5 n 5 8.

tor

we have the following D(n)'s:

n=5, if

n=6,

(3,3,3,3,3,3,3), (2,2,2,2,4,5,4), (2,4,5,4,4,5,4)

if

n=7

(3,2,2,2,5,6,6,6), (4,3,3,3,4,5,5,5), (2,4,6,3,6,4,2,5), (4,3,6,4,2,3,4,6), (4,2,3,2,2,2,3,6), (4,3,2,5,5,2,5,6), (4,2,4,3,4,6,4,5), (4,5,5,5,2,2,3,6), (4,5,6,6,5,4,5,5),

n=S.

(2,3,2,5,2,3,2,5), (2,3,2,5,5,5,5,5), if It can be seen without difficulty that

(Cn,n)

corresponding to these

D(~r)'s

excepting n = 5 are hamiltonian. This result was also obtained in [ S ] using computer. The permutation IT n

k,n

(i)

=

is given by k,n residue of ik(mod n)

if

1

i 5 n-1

5

and

n

k,n

(n) = n ,

) of k and n are coprime and 1 5 k 5 n / 2 . The subfamily (cn, n k,n generalized n-prisms is isomorphic to the generalized Petersen graphs G(n,k)

where for

(n,k)

=

1.

It was shown by Robertson [9] that the graphs if and only if

n

z

5(mod 6).

G(n,2)

The nonhamiltonian G(n,2)

are non-hamiltonian graphs are now known

as the Robertson graphs. The result that the graphs G(n,3) , (n,3) hamiltonian except for the Petersen graph

(n=5)

=

is due to Bondy [3].

1 are Kozo

Bannai [l] has obtained the following. THEOREM 4. Ccficralized Petersen graphs

G(n,k)

with

(n,k)

=

1 are

hamiltonian unless t h e y a r e isomorphic t o Robertson graphs. As

G(n,k)

has

D(rr

)

k,n a constant sequence, i . e . , if

which we must have k = 2

when

(n,k)

=

=

(k,k,...k)

D(n)

1 , then

we have one more result on

is a constant sequence (Cn,n)

(k,k,...k)

D(n), for

is always hamiltonian excepting

nes(mod6).

The anlysis could not proceed as we could not obtain any general result on D(n)

necessary for o u r purpose. ACKNOWLEDGEMENT The authors are extremely thankful to the referee for his helpful comments.

214

S.P. Mohanty and D.Rao REFERENCES Bannai, Kozo, Hamiltonian cycles in generalized Paterson graph, J. Combinatorial Theory Ser. B, 24 (1978), 181-188. Behzad, M. and Chartrand, G., Introduction to the Theory of Graphs, (Allyn and Bacon, Boxton 1971).

131 Bondy, J.A., Variations on the Hamiltonian theme, Can. Math. Bull. 15 (19721, 57-62. [41 Chartrand, G. and Harary, F., Planar permutation graphs, Ann. Inst. Henin Poincare, Vol. 11 No. 4 (19671, 433-438. Harary, F., Graph Theory, (Addison-Wesley, Reading, Mass., 1969). Hedetniemi, S . , On Classes of Graphs Defined by Special Cutsets of Lines in the Many Facets of Graph Theory, Springer Verlag, Lecture Notes in Mathematics, No. 110, 171-190. [71 Holton, D.A. and Stacey, K.C., Some Problems in Permutation Graphs, School of Mathematical Sciences Research Report No. 18, University o f Melbourne, Melbourne, (1974). Klee, V., Which Generalized Prisms Admit H-circuits, Graph Theory and Applications, (Y. Alavi, D.R. Lick and A.T. White, eds.) SpringerVerlag, Lecture Notes in Mathematics, No. 303, 173-179. Robertson, G.N., Graphs under Girth, Valency and Connectivity Constraints, (Dissertation), University of Waterloo, Waterloo, Ontario, Canada, 1968.