4 DECOMPOSITION OF COMPLETE GRAPHS INTO CIRCULANT...

International Journal of Computer Engineering and Technology (IJCET), ISSN 0976-6367(Print),

ISSN 0976 - 6375(Online), Volume 4, Issue 6, November - December (2013), © IAEME

25

DECOMPOSITION OF COMPLETE GRAPHS INTO CIRCULANT GRAPHS

AND ITS APPLICATION

1Jayanta Kr. Choudhury,

2Anupam Dutta,

3Bichitra Kalita

1Swadeshi Academy Junior College, Guwahati-781005

2Patidarrang College, Muktapur, Loch, Kamrup, Assam

3Department of Computer Application, Assam Engineering College, Guwahati-781013

ABSTRACT

In this paper, the decomposition of complete graphs 12 +mK for 2≥m into circulant graphs has

been discussed. Two theorems have been established for different values of 2≥m relating to 12 +m

is prime, n

m 3312 ⋅=+ for 1≥n and sm 312 =+ for 5≥s is also prime. Finally, an algorithm under

different situation for the traveling salesman problem have been discussed when the weights of edges

are non-repeated of the complete graph 12 +mK for 2≥m .

Keywords: Algorithm, Circulant Graphs, Complete Graphs, Hamiltonian Graphs,

Traveling Salesman Problem (TSP).

1. INTRODUCTION

Circulant graph have many applications in areas like telecommunication network, VLSI

design and distributed computing. Circulant graph is a natural extension of a ring, with increased

connectivity. Zbgniew R. Bogdanowicz [1] give the necessary and sufficient conditions under which

a directed circulant graph G of order n and with k jumps can be decomposed into k pair wise arc-

disjoint anti-directed Hamiltonian cycles, each induced by two jumps. In addition, the necessary

condition for complete decomposition of G into arbitrary anti-directed Hamiltonian cycles has been

discussed. Matthew Dean [2] shown that there is a Hamiltonian cycle decomposition of every 6-

regular circulant graph n

S in which S has an element of order n , { }0\nZS ⊆ has vertex set nZ

and edge set [ ]{ }SsZxsxx n ∈∈+ ,|, . S. El-Zanati, Kyle King and Jeff Mudrock [3] discussed the

cyclic decomposition of circulant graphs into almost bipartite graphs. Dalibor Froncek [4] used

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ISSN 0976 – 6367(Print)

ISSN 0976 – 6375(Online)

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26

labeling to show the cyclic decomposition of complete graphs into eK nm +, : the missing case. J.C.

Bermond, O. Favaron and M. Maheo [7] prove that any 4-regular connected Cayley graph on a finite

abelian group can be decomposed into two Hamiltonian cycles. Daniel K. Biss [5] shown that the

circulant graph ( )dcdGm , is Hamiltonian decomposable for all positive integers dc, and m with

dc < where the graph ( )dNG , has vertex set { }1.,,.........2,1,0 −= NV , with { }wv, an edge if

( )Ndwv mod±≡− for some 1log0 −≤≤ Ni d . This extends work of Michenean [6]. Marsha F.

Foregger [8] settles Kotzig’s [9] conjecture that the Cartesian product of any three cycles can be

decomposed into three Hamiltonian cycles. He also show that the Cartesian product of ba

32 graphs,

each decomposed into mba32 Hamiltonian cycles. Dave Morris [10] discussed the status of the

search for Hamiltonian cycles in circulant graphs and circulant digraphs. R. Anitha and R. S. Lekshmi

[13] have proved that the complete graph nK2 can be decomposed into 2−n n -suns, a Hamiltonian

cycle and a perfect matching, when n is even and for odd cases, the decomposition is 1−n n -suns

and a perfect matching. They also discussed that a spanning tree decomposition of even order

complete graph using the labeling scheme of n -suns decomposition. A complete bipartite graph nnK ,

can be decomposed into 2

n n -suns when

2

n is even. When

2

n is odd, nnK , can be decomposed into

( )2

2−n n -suns and a Hamiltonian circuit [13]. Tay-Woei Shyu [14] investigated the decomposition of

complete graph nK cycles tC ’ s and stars kS ’ s and studied necessary or sufficient conditions for

such a decomposition to exist. Dalibor Froncek [15] show that every bipartite graph H which

decomposes kK and nK also decomposes knK . Darryn E. Bryant and Peter Adams [16] discussed

that for all 3≥v , the complete graph vK on v vertices can be decomposed into 2−v edge disjoint

cycles whose lengths are 1,,5,4,3,3 −vKK and for all 7≥v , vK can be decomposed into 3−v edge

disjoint cycles whose lengths are vvvv ,1,2,4,,4,3 −−−KK . Fu, Hwang, Jimbo, Mutoh and Shiue

[17] considered the problem of decomposing a complete graph into the Cartesian product of two

complete graphs rK and cK and they found a general method of constructing such decomposition

using various sorts of combinatorial designs. The traveling salesman problem (TSP) in the circulant

weighted undirected graph case have been discussed by I. Gerace and F. Greco [11]. They discussed

an upper bound and a lower bound for the Hamiltonian case and analyzed the two stripe case.

Moreover, I. Gerace and F. Greco [12] studied short SCTSP (symmetric circulant traveling salesman

problem) and they presented an upper bound, a lower bound and a polynomial time 2-approximation

algorithm for the general case of SCTSP (symmetric circulant traveling salesman problem). Further,

different types of algorithms and results have also been focused by B.Kalita [19-22], J.K. choudhury

[23-24] and A. Dutta[25-26].

In this paper, we present a theorem for decomposition of complete graphs 12 +mK into m edge

disjoint circulant graphs ( )jC m 12 + for 2≥m where j is the jump, mj ≤≤1 . Thereafter, we present

a theorem for the decomposition of 12 +mK into edge disjoint circulant graphs whenever 12 +m is a

prime, 1,3312 ≥⋅=+ nmn

and sm 312 =+ , 5≥s is a prime and this circulant graphs lead to

determine edge disjoint Hamiltonian cycle. Finally, an algorithm has been developed to determine the

least cost route of a traveler.



27

The paper is organized as follows The section 1 includes the introduction part containing works of other researcher. Section 2

includes notations and terminologies. In section 3, two theorems are stated and proved. Section 4

includes an algorithm. Section 5 explains experimental result and the conclusion is included in

section 6.

2. NOTATION AND TERMINOLOGY

The notation and terminologies have been considered from the standard references [1-29]

Definition 2.1[18] (circulant matrix): Every nn × matrix C of the form

=

−

−

−−

−−

0121

12

01

2101

1210

cccc

cc

cc

cccc

cccc

C

n

n

nn

nn

L

OO

MOM

L

is called a circulant matrix. The matrix C is completely determined by its first row because other

rows are rotations of the first row. C is symmetric if iin cc =− for 1,.......,2,1 −= ni . Further, C is

an adjacency matrix if 00 =c and { }1,01 ∈=− in cc .

Definition 2.2[18] (circulant graph): A circulant graph is a graph which has a circulant adjacency

matrix.

Examples of circulant graphs are the cycle nC , the complete graph nK , and the complete

bipartite graph nnK , .

Circulant Graph 2.3 [27] : For a given positive integer, let knnn ,,, 21 KK be a sequence of integers

where

( )2

10 21

+<<<<<

pnnn kKK .

Then the circulant graph ( )kp nnnC ,,, 21 KK is the graph on p nodes pvvv ,,, 21 KK with vertex

iv adjacent to each vertex ( )pni jv mod± . The values in are called jump sizes.

Definition 2.4[28]: A graph G is said to be decomposable into the sub

graphs iHHHH ,,,, 321 KK , ki ≤≤1 , having no isolated vertices and

( ) ( ) ( ) ( ){ }kHEHEHEHE ,,,, 321 KK is a partition of ( )GE .

3. THEOREMS

3.1 Theorem: The complete graph 12 +mK for 2≥m can be decomposed to m edge disjoint circulant

graph ( )jC m 12 + where j is the jump and mj ≤≤1 .



28

Proof: Consider the complete graph 1225 +×= KK . Then vertex set of 5K is { }54321 ,,,, vvvvvV = and

it has ( )

102

45

2

1=

×=

−nn edges. Let us partitioned the edges of 5K

into 2 distinct sets containing

5 edges each as { }15544332211 ,,,, vvvvvvvvvvE = and { }25145342312 ,,,, vvvvvvvvvvE = respectively.

Adjoining edges of 1E and 2E , we can construct 2 edge disjoint circulant graphs ( )15C and ( )25C

for the jumps 1=j and 2=j respectively and their respective permutations can be represented as 1f

and 2f where

=

15432

54321

1vvvvv

vvvvvf and

=

21543

54321

2vvvvv

vvvvvf

i.e., ( )543211 vvvvvf = and ( )425312 vvvvvf = . Figure-1 shows two circulant graphs ( )15C and ( )25C

for the complete graph 5K .

( )15C ( )25C

Figure-1

Now, we append the edges to the circulant graphs as

( )

=

≤≤=

−

+

5

41

4 iifv

iifvvf

i

ji

ij , 1=j

and ( )

≤≤

≤≤=

−

+

54

31

3 iifv

iifvvf

i

ji

ij , 2=j

Similarly, consider the complete graph 1327 +×= KK whose vertex set is { }7654321 ,,,,,, vvvvvvvV =

and have ( )

212

67

2

1=

×=

−nn edges. Let us partitioned the edges of 7K into 3 distinct sets

consisting of 7 edges each, as { }177665544332211 ,,,,,, vvvvvvvvvvvvvvE = ,

{ }271675645342312 ,,,,,, vvvvvvvvvvvvvvE = and { }372615746352413 ,,,,,, vvvvvvvvvvvvvvE = . Adjoining

7 edges each of 1E , 2E and 3E , we can construct 3 edge disjoint circulant graphs ( )17C , ( )27C and

( )37C for the jumps 1=j , 2=j and 3=j respectively and their respective permutations are 1f , 2f

and 3f where



29

=

1765432

7654321

1vvvvvvv

vvvvvvvf ,

=

2176543

7654321

2vvvvvvv

vvvvvvvf and

=

3217654

7654321

3vvvvvvv

vvvvvvvf

i.e., ( )76543211 vvvvvvvf = , ( )64275312 vvvvvvvf = and ( )52637413 vvvvvvvf = . Figure-2 shows circulant

graphs ( )17C , ( )27C and ( )37C .

v

1

v2

v3

v4

v5

v6

v7

v1

v2

v3

v4 v

5

v6

v7

v1

v2

v3

v4

v5

v6

v7

( )17C ( )27C ( )37C

Figure-2

The edges to the circulant graphs can be appended as

( )

=

≤≤=

−

+

7

61

6 iifv

iifvvf

i

ji

ij , 1=j

( )

≤≤

≤≤=

−

+

76

51

5 iifv

iifvvf

i

ji

ij , 2=j

and ( )

≤≤

≤≤=

−

+

75

41

4 iifv

iifvvf

i

ji

ij , 3=j

Similarly, partitioning the edges of 9K into 4 distinct sets containing 9 edges each as

{ }1998877665544332211 ,,,,,,,, vvvvvvvvvvvvvvvvvvE = ,


{ }3928179685746352413 ,,,,,,,, vvvvvvvvvvvvvvvvvvE =

and { }4938271695847362514 ,,,,,,,, vvvvvvvvvvvvvvvvvvE =

respectively, we can construct 4 edge disjoint circulant graphs ( )19C , ( )29C , ( )39C and ( )49C for the

jumps 1=j , 2=j , 3=j and 4=j respectively and their corresponding permutations are given by



30

( )987654321

198765432

987654321

1 vvvvvvvvvvvvvvvvvv

vvvvvvvvvf =

= ,

( )864297531

219876543

987654321


vvvvvvvvvf =

= ,

( )( )( )963852741

321987654

987654321


vvvvvvvvvf =

=

and ( )627384951

432198765

987654321


vvvvvvvvvf =

=

respectively, whose structures are shown in Figure-3.

V

1

V2

V3

V4

V5 V

6

V7

V8

V9

V1

V2

V3

V4

V5 V

6

V7

V8

V9

V1

V2

V3

V4

V5 V6

V7

V8

V9

( )19C ( )29C ( )39C ( )49C

Figure-3

Append the edges to the circulant graphs as

( )

=

≤≤=

−

+

9

81

8 iifv

iifvvf

i

ji

ij , 1=j

( )

≤≤

≤≤=

−

+

98

71

7 iifv

iifvvf

i

ji

ij , 2=j

( )

≤≤

≤≤=

−

+

97

61

6 iifv

iifvvf

i

ji

ij , 3=j

and ( )

≤≤

≤≤=

−

+

96

51

5 iifv

iifvvf

i

ji

ij , 4=j

V1

V2

V3

V4

V5 V6

V7

V8

V9

V1

V2

V3

V4

V5 V

6

V7

V8

V9



31

From the above discussion, it is seen that the complete graph 12 +mK for 2≥m can be

decomposed into m edge disjoint circulant graph ( )jC m 12 + . Here j represents the jump of the

circulant also depends upon m . If 2=m , then 2,1=j and we have 2 circulant graphs ( )15C and

( )25C .

Again, if 3=m , then 3,2,1=j and then 7K decomposes into 3 edge disjoint circulant

graphs ( )17C , ( )27C and ( )37C and so on. Moreover, in their respective permutations, the images of

first element and last element in the first row are respectively 1+jv and jv . Here we excluded the

identity permutations as well as permutations which produce the same circulant graphs,

i.e.

15432

54321

vvvvv

vvvvv and

43215

54321

vvvvv

vvvvv represents the same circulant graph ( )15C .

Therefore, as discussed above, we get 2 permutations if 2=m , 3 permutations if 3=m etc, i.e.,

number of permutations are depends upon the value of 2≥m .

Finally, we can comment that the complete graph 12 +mK for 2≥m can be decomposed into

m edge disjoint circulant graph ( )jC m 12 + for the jump mj ≤≤1 and their corresponding m

permutations are defined as

( )( )

( ) ( )

+≤≤−−

−−≤≤=

−−+

+

1222

121

21 mijmifv

jmiifvvf

mji

ji

ij

for mj ≤≤1 and which complete the theorem.

Note: Here, complete graphs 12 +mK for 2≥m has odd number of vertices and therefore they play

different role for different values of 2≥m . In the next theorem we discuss only three cases.

3.2 Theorem: Let 12 +mK be a complete graph for 2≥m . Then following conditions are satisfied

(a) If 12 += mp is a prime number, then pK can be decomposed into m edge disjoint circulant

graphs ( )jC p where j is the jump and mj ≤≤1 .

(b) If nmq 3312 ⋅=+= where ( )13

2

1 1 −= +nm for 1≥n , then qK can be decomposed into

(i) n3 edge disjoint circulant graphs ( )jCq for the jump mj ≤≤1 but lj 3≠ for

( )132

11 −≤≤ n

l and

(ii) a circulant graph ( )lCq 3,,9,6,3 KK for the jump l3 where ( )132

11 −≤≤ n

l .

(c) If smr 312 =+= where 5≥s is a prime number, then rK can be decomposed into

(i) 1−s edge disjoint circulant graphs ( )jCr for the jump mj ≤≤1 where sj ≠ and ej 3≠ ,

( )13

11 −≤≤ me

(ii) a circulant graph ( )eCr 3,,9,6,3 KK for the jump ( )13

11 −≤≤ me and

(iii) a circulant graph ( )sCr for the jump s .



32

Proof: (a) When =+= 12mp prime, then it follows that the complete graph 5K , 7K , 11K ,

13K ,……….. for KKK,6,5,3,2=m exist and the proof is same as discussed in Theorem 3.1. Hence

the complete graph pK where 12 += mp is a prime for 2≥m can be decomposed into ( )12

1−= pm

edge disjoint circulant graphs ( )jC p for the jump mj ≤≤1 .

In the above discussion, we observed that each edge disjoint circulant graph ( )jC p for the

jump ( )12

11 −≤≤ pj is regular of degree two. In other words, they are cycle pC of length p and

therefore they represent edge disjoint Hamiltonian circuits. So in this discussion, number of edge

disjoint Hamiltonian circuits in the complete graph pK is m if 12 += mp is a prime for 2≥m .

Proof: (b) Let 12 +mK be a complete graph for 2≥m . Suppose that n

mq 3312 ⋅=+= such that

( )132

1 1 −= +nm for 1≥n . Then { }qvvvvV ..,,.........,, 321= is the vertex set of qK and it has ( )

2

1−qq edges.

Let us define ( ) ( )qqj KVKVf →: as follows:

(i) when ( )132

11 1 −≤≤ +n

j but lj 3≠ for ( )132

11 −≤≤ n

l 1≥∀n

( )

≤≤+−

−≤≤=

−+

+

qijqifv

jqiifvvf

qji

ji

ij1

1

and (ii) when ( )132

11 −≤≤ n

l 1≥∀n , ( )

≤≤+−

−≤≤=

−+

+

qilqifv

lqiifvvf

qli

li

il13

31

3

3

3

For 1=n , 4=m and 1339142 ⋅==+×=q . Then { }987654321 ,,,,,.,, vvvvvvvvv is the vertex set of

9K have ( )

36942

89

2

1=×=

×=

−qq edges. Since 4=m , we can partition the edges of 9K into 4 edge

disjoint sets, each set containing 9 edges, by the above rule and their permutations are given as

follows.

For 1=j , ( )987654321

198765432

987654321


vvvvvvvvvf =

=

For 2=j , ( )864297531

219876543

987654321


vvvvvvvvvf =

=

For 4=j , ( )627384951

432198765

987654321


vvvvvvvvvf =

=

Again, for 1=l , ( )( )( )963852741

321987654

987654321


vvvvvvvvvf =

=



33

From these permutations, we get 4 edge disjoint sets



{ }3928179685746352413 ,,,,,,,, vvvvvvvvvvvvvvvvvvE = and

{ }4938271695847362514 ,,,,,,,, vvvvvvvvvvvvvvvvvvE = .

Adjoining edges of 1E , 2E and 4E , we get 331 = edge disjoint circulant graphs ( )19C ,

( )29C and ( )49C for the jumps 1=j , 2=j and 4=j respectively [Figure-3]. Also, adjoining the

edges of 3E , we get a circulant graph ( )39C for the jump l3 where 1=l . Thus 9K decomposed into

13 edge disjoint circulant graphs ( )jC9 for 41 ≤≤ j , 3≠j and a circulant graph ( )lC 39 for 1=l

[Figure-3].

For 2=n , 13=m and 233271132 ⋅==+×=q . Then { }27321 ,,, vvvv KKK is the vertex set of the

complete graph 27K and have ( )3512713

2

2627

2

1=×=

×=

−qq edges. Since 13=m , we can

partition the edges of 27K into 13 edge disjoint sets, each set containing 27 edges, by the rule

defined above.

For 1=j , ( )2726252423222120191817161514131211109876543211 vvvvvvvvvvvvvvvvvvvvvvvvvvvf =









Again,

For 1=l , ( )( )( )2724211815129632623201714118522522191613107413 vvvvvvvvvvvvvvvvvvvvvvvvvvvf =


For 3=l , ( )( )( )( )( )( )( )( )( )2718926178251672415623145221342112320112191019 vvvvvvvvvvvvvvvvvvvvvvvvvvvf =




34

From these permutations we can define corresponding edge disjoint sets 1E , 2E , 3E , 4E , 5E ,

6E , 7E , 8E , 9E , 10E , 11E , 12E and 13E respectively.

Adjoining edges of 1E , 2E , 4E , 5E , 7E , 8E , 10E , 11E and 13E , we can construct 932 = edge

disjoint circulant graphs ( )127C , ( )227C , ( )427C , ( )527C , ( )727C , ( )827C . ( )1027C , ( )1127C and

( )1327C for the jumps 1=j , 2=j , 4=j , 5=j , 7=j , 8=j , 10=j , 11=j and 13=j

respectively. Also, adjoining edges of 4E , 6E , 9E and 12E , we get a circulant graph ( )12,9,6,327C

for the jump l3 where 41 ≤≤ l . Thus the complete graph 27K can be decomposed into 932 =

edge disjoint circulant graph ( )jC27 where j is the jump such that 131 ≤≤ j , 12,9,6,3≠j and a

circulant graph ( )12,9,6,327C .

Let kn = so that ( )132

1 1 −= +km and

133312 +=⋅=+= kkmq . Then { }qvvvv ,,,, 321 KKK is

the vertex set the complete graph 13 += kKK q and it has ( ) ( )2

133

2

1 11 −=

− ++ kkqq edges.

Since ( )132

1 1 −= +km , so we can partition edges of 13 += kKK q into ( )132

1 1−

+k distinct sets,

each set containing 13 +k edges, given by

(i) when ( )132

11 1 −≤≤ +kj but lj 3≠ for ( )13

2

11 −≤≤

kl

( )

≤≤+−

−≤≤=

−+

+

qijqifv

jqiifvvf

qji

ji

ij1

1


11 −≤≤ kl

( )

≤≤+−

−≤≤=

−+

+

qilqifv

lqiifvvf

qli

li

il13

31

3

3

3

Now, we can have edge disjoint sets 1E , 2E , 3E , 4E ,……………,( )13

2

1 1−+kE and their corresponding

permutations are 1f , 2f , 3f , 4f ,…………….,( )13

2

1 1−+kf .

Adjoining edges of 1E , 2E , 4E , 5E , 7E ,……, ( )53

2

1 1 −+kE ,

( )132

1 1 −+kE , we can construct

k3 edge

disjoint circulant graphs ( )jC k 13 + where j is the jump and ( )132

11 1 −≤≤ +k

j , lj 3≠ for

( )132

11 −≤≤ k

l . Also, adjoining edges of 3E , 6E , 9E ,……,( )73

2

1 1 −+kE ,

( )332

1 1 −+kE , we get a circulant

graph ( )lC k 3,,9,6,313KK+ for the jump ( )13

2

11 −≤≤ kl .

Now, adding 132 +⋅ k additional vertices to the vertex set and ( )11 394 ++ −⋅ kk

additional edges to the

edge set of the complete graph 13 +kK . Then { }2221 313133321 ,,,,,,,, ++++ −+ kkkk vvvvvvv KK will be the vertex

set for the complete graph 12 333 ++ ⋅= kk KK i.e., 23 += kKK q and

( ) ( ){ }132

113

2

13123 11222 −=−=⇒=+⇒= +++++ kkkk mmq .



35

The complete graph 23 +kK has ( ) ( ) ( ) ( ){ }

2

133

2

133

2

1111122

−=

−=

−++++++ kkkk

qq edges. These edges can be

partition into ( ) ( ){ }132

113

2

1 112 −=− +++ kk disjoint sets and their respective permutations are given by

(i) when ( )132

11 2 −≤≤ +kj but lj 3≠ for ( )13

2

11 1 −≤≤ +kl

( )

≤≤+−

−≤≤=

−+

+

qijqifv

jqiifvvf

qji

ji

ij1

1


11

1−≤≤

+kl , ( )

≤≤+−

−≤≤=

−+

+

qilqifv

lqiifvvf

qli

li

il13

31

3

3

3

Now, we can have ( ){ }13

2

1 11 −++k edge disjoint sets 1E , 2E , 3E , 4E , …,

( )132

1 2 −+kE and their

respective permutations are 1f , 2f , 3f , 4f ,………,( )13

2

1 2 −+kf . Adjoining edges of 1E , 2E , 4E , 5E ,

7E ,……..,( )73

2

1 2 −+kE ,

( )532

1 2 −+kE ,

( )132

1 2 −+kE , we can construct

13 +k edge disjoint circulant graph

( )jC k 23 + i.e., ( ) ( )jC k 113 ++ where j is the jump and ( )( )132

11 11 −≤≤ ++k

j , lj 3≠ for ( )132

11 1 −≤≤ +kl .

Also, adjoining edges of 3E , 6E , 9E ,….., ( )93

2

1 2 −+kE ,

( )332

1 2 −+kE , we have a circulant graph

( ) ( )lC k 3,,9,6,3113KK++ for the jump ( )13

2

11 1 −≤≤ +k

l . Thus the result is true for 1+= kn whenever

it is true for kn = . Hence by principle of mathematical induction, the result is true for all Nn ∈ .

In this discussion, we observed that each edge disjoint circulant graph ( )jCq

for the jump

( )132

11 1 −≤≤ +n

j , lj 3≠ for ( )132

11 −≤≤ n

l are regular of degree two. In other words they are

cycle qC of length q and therefore they represent n3 edge disjoint Hamiltonian circuits 1≥n .

Further the circulant graph ( )lC q 3,,9,6,3 KK for the jump ( )132

11 −≤≤

nl , as we have observed,

can be decomposed into ( )132

1−n edge disjoint circulant graphs ( )tCq 3 for the jump t3 ,

( )132

11 −≤≤ n

t for all 1≥n . If the jump is

nt 33 = , then

n3 3K -decomposition

133 −= n

t , then n3 9C -decomposition

233 −= nt , then

n3 27C -decomposition

………………………………………..

………………………………………..

333 =t , then

n3 23 −nC -decomposition

233 =t , then

n3 13 −nC -decomposition

133 =t , then

n3 nC3

-decomposition



36

Clearly above cycles does not represent Hamiltonian circuits. So in this discussion, number of edge

disjoint Hamiltonian circuits in the complete graph qK is n if

nq 33⋅= for 1≥n .

(c) Let 12 +mK be a complete graph for all 2≥m . Suppose that smr 312 =+= where 5≥s is a

prime number and 2

13 −=

sm . Then { }rvvvvV ..,,.........,, 321= is the vertex set of the complete graph

rK i.e., sK3 and have ( )2

1−rr edges.

Let us define ( ) ( )ssj KVKVf →: as follows:

(i) when ( )132

11 −≤≤ sj but sj ≠ and ej 3≠ for ( )1

2

11 −≤≤ se

( )

≤≤+−

−≤≤=

−+

+

rijrifv

jriifvvf

rji

ji

ij1

1

(ii) when ( )12

11 −≤≤ se , ( )

≤≤+−

−≤≤=

−+

+

rierifv

eriifvvf

rei

ei

ie13

31

3

3

3

and (iii) ( )

≤≤+−

−≤≤=

−+

+

risrifv

sriifvvf

rsi

si

is1

1

For 5=s , a prime number, we have 1553 =×=r and 72

153=

−×=m .

Then { }1514321 ,..,,.........,, vvvvv is the vertex set of the complete graph 15K and have

( )105157

2

1415

2

1=×=

×=

−rredges. Since 7=m , we can partition the edges of 15K into 7 disjoint

sets, each set containing 15 edges and their corresponding permutations by the above rule are as

follows:

For 1=j , ( )1514131211109876543211 vvvvvvvvvvvvvvvf = and

{ }1151514141313121211111010998877665544332211 ,,,,,,,,,,,,,, vvvvvvvvvvvvvvvvvvvvvvvvvvvvvvE =







For 1=e , ( )( )( )1512963141185213107413 vvvvvvvvvvvvvvvf = and


For 2=e , ( )( )( )1261593115148210413716 vvvvvvvvvvvvvvvf = and


For 5=s , ( )( )( )( )( )1510514941383127211615 vvvvvvvvvvvvvvvf = and




37

Adjoining edges of 1E , 2E , 4E and 7E , we can construct 4151 =−=−s edge disjoint circulant

graph ( )115C , ( )215C , ( )415C and ( )715C respectively for jumps 1=j , 2=j , 4=j

and 7=j [Figure-5]. Also, adjoining edges of 3E and 6E , we can construct a circulant graph

( )6,315C [Figure-6]. Again, adjoining the edges of 5E , we can construct a circulant graph ( )515C for

the jump 5=s [Figure-7]. V

1

V2

V3

V5

V6

V7

V8V

9

V11

V10

V12

V13

V14

V15

V4

V1

V2

V3

V5

V6

V7

V8

V9

V11

V10

V12

V13

V14

V15

V4

( )115C ( )215C ( )415C ( )715C

Figure-5

V1

V2

V3

V5

V6

V7

V8 V

9

V11

V10

V12

V13

V14

V15

V4

V1

V2

V3

V5

V6

V7

V8V

9

V11

V10

V12

V13

V14

V15

V4

( )6,315C ( )515C

Figure-6 Figure-7

Thus the complete graph 15K can be decomposed into 4151 =−=−s edge disjoint circulant

graphs ( )jC15 for 71 ≤≤ j , 6,5,3≠j and a circulant graph ( )6,315C and a circulant graph ( )515C

for 5=s . Similarly, we can show for KK,13,11,7=s etc.

Let ks = , a prime number. Then kr 3= , 2

13 −=

km and the complete graph kK3 has

( ) ( )2

133

2

1 −=

− kkrr edges. Since 2

13 −=

km , so we can partition the edges of kK3 into ( )13

2

1−k

edge disjoint sets, each set containing k3 edges and their respective permutations are given by

(i) when ( )132

11 −≤≤ kj but kj ≠ and ej 3≠ for ( )1

2

11 −≤≤ ke

( )

≤≤+−

−≤≤=

−+

+

rijrifv

jriifvvf

rji

ji

ij1

1

(ii) when ( )12

11 −≤≤ ke , ( )

≤≤+−

−≤≤=

−+

+

rierifv

eriifvvf

rei

ei

ie13

31

3

3

3

and (iii) ( )

≤≤+−

−≤≤=

−+

+

rikrifv

kriifvvf

rki

ki

ik1

1

V1

V2

V3

V5

V6

V7

V8V

9

V11

V10

V12

V13

V14

V15

V4

V1V

2

V3

V5

V6

V7

V8

V9

V11

V10

V12

V13

V14

V15

V4



38

Now, we can have edge disjoint sets 1E , 2E , 3E , 4E ,……,( )13

2

1−k

E and their respective permutations

are 1f , 2f , 3f , 4f ,………,( )13

2

1−k

f . Adjoining edges of jE ’s where ( )132

11 −≤≤ kj ; kj ≠ and

ej 3≠ for ( )12

11 −≤≤ ke , we can construct 11 −=− ks edge disjoint circulant graphs ( )jC k3

where j is the jump. Again, adjoining the edges of eE3 ’s, ( )12

11 −≤≤ ke we can construct a

circulant graph ( )eC k 3,,9,6,33 KK for the jump e3 . Also, adjoining the edges of kE , we can

construct a circulant graph ( )kC k3 for the jump k .

In this discussion we observed that each edge disjoint circulant graphs ( )jCr for the jump

( )132

11 −≤≤ sj , sj ≠ and ej 3≠ for ( )1

2

11 −≤≤ se where 5≥s is a prime, are regular of degree

two. In other words they are cycles rC of length r and therefore they represent edge disjoint

Hamiltonian circuits. Again the circulant graph ( )eCr 3,,9,6,3 KK for the jump e3 where

( )12

11 −≤≤ se can be decomposed into ( )sm − edge disjoint circulant graphs t3 where

( )12

11 −≤≤ st and these circulants are regular of degree two each and they can be further

decomposed into ( )1−m cycles sC of length rs < and so they do not represent Hamiltonian

circuits. Also, the circulant graph ( )sCr for the jump s is regular of degree two and from which we

can have s 3K -decompositions and they do not represent a Hamiltonian circuit. So in this discussion,

number of edge disjoint Hamiltonian circuits in the complete graph rK is 1−s if sr 3= , 5≥s is a

prime.

4. ALGORITHM

Input: Let 12 +mK for 2≥m be a complete graph.

Output: Find least cost route.

Case 1: When 12 += mp is a prime number and ( )12 +m consecutive greatest weights are incident

at different vertices of the complete graph 12 +mK for 2≥m along edges of a cycle 12 +mC , next

( )12 +m greatest weights incident at different vertices along edges of a cycle 12 +mC other than the

previous one, next ( )12 +m greatest weights incident at different vertices along edges of a cycle

12 +mC other than the previous two cycles and so on.

Step 1: Study the graph 12 +mK for 2≥m .

Step 2: Delete ( )12 +m number of consecutive greatest weights incident at different

vertices along the edges of a cycle 12 +mC of the complete graph 12 +mK for 2≥m .

Step 3: Observed that the graph obtained in Step 2 is a regular sub graph of the complete graph

12 +mK of degree 2 . Otherwise go to Step 6.

Step 4: Observed that the graph obtained in Step 3 is a cycle 12 +mC for 2≥m attached with

minimum weights and this is the required Hamiltonian circuit i.e., least cost route.

Step 5: Stop



39

Step 6: From the graph obtained in Step 2, delete another ( )12 +m number of edges attached with

consecutive greatest weights other than ( )12 +m consecutive greatest weights already deleted in

Step 2.

Step 7: Go to Step 3 to Step 5. Otherwise go to Step 8.

Step 8: Repeat Step 6 till reaching the stage of Step 3 to Step 4 and go to Step 5.

Case 2: When nmq 3312 ⋅=+= for 1≥n and

13 +n number of consecutive greatest weights incident

at different vertices along edges of n3 edge disjoint 3K , next

23 +n number of consecutive greatest

weights attached with edges of n3 edge disjoint cycles 23

C , next 33 +n number of consecutive greatest

weights attached with edges of n3 edge disjoint cycles 33

C and so on the remaining 123 +n number of

consecutive greatest weights are attached with edges of n3 circuits 13 +nC .

Step 9: Delete 13 +n number of consecutive greatest weights incident at different vertices along edges

of n3 edge disjoint 3K .


nK33⋅

of degree n32 ⋅ for 1≥n .

Step 11: From the graph obtained in Step 11, delete another 13 +n number of edges forming the

circuit 13 +nC attached with consecutive greatest weights other than 13 +n consecutive greatest weights

already deleted in Step 9, successively ( )13 −n times.


nK33⋅

of degree 2 for 1≥n . Otherwise go to Step 15.

Step 13: Observed that graph obtained in Step 12 is a circuit 13 +nC attached with minimum weights,

which is the required least cost route.

Step 14: Go to Step 5.


nK33⋅

of degree ( )133 1 −+n for 1≥n .

Step 16: From the graph obtained in Step 15, delete another 23 +n number of consecutive greatest

weights other than the greatest weights already deleted in Step 11, incident at different vertices

forming n3 edge disjoint cycles 23

C for 1≥n .





C for 1≥n .





C for 1≥n and continue this process till the graph becomes

regular of degree n32 ⋅ for 1≥n .

Step 21: Go to Step 10 to Step 14.



40

Case 3: When smr 312 =+= , 5≥s is a prime and s3 number of consecutive greatest weights

incident at different vertices forming s edge disjoint 3K , next ( )sms −3 number of consecutive

greatest weights incident at different vertices forming ( )sm −3 edge disjoint cycles sC and the

remaining ( )13 −ss number of consecutive greatest weights incident at different vertices forming

( )sm − edge disjoint cycles sC3 .

Step 22: Delete s3 number of consecutive greatest weights incident at different vertices forming s

edge disjoint 3K for 5≥s .


sK3 of degree ( )13 −s for 5≥s .

Step 24: From the graph obtained in Step 23, delete another ( )sms −3 number of edges forming

( )sm −3 edge disjoint circuits sC attached with greatest consecutive weights, other than s3 number

of consecutive greatest weights already deleted in Step 22.


sK3 of degree ( )12 −s having ( )13 −ss number of remaining edges for 5≥s .

Step 26: From the graph obtained in Step 25, delete another s3 number of edges forming the cycle

sC3 attached with consecutive greatest weights which are different from greatest weights already

deleted in Step 24.


sK3 of degree ( )22 −s for 5≥s .

Step 28: Repeat Step 26 on the graph obtained in Step 23 for ( )3−s times where 5≥s .

Step 29: Observed that the graph obtained in Step 28 is a cycle sC3 attached with minimum weights

and this is the required Hamiltonian circuit i.e. , least cost route.

Step 30: Go to Step 5

5. EXPERIMENTAL RESULTS FOR ALGORITHM

Case 1: The following cost matrix (Table-1) is considered for seven cities (i.e., for 3=m of 12 +mK

and 712 =+m , a prime)

A B C D E F G

A ∞ 20 3 35 51 18 34

B 20 ∞ 23 13 47 43 10

C 3 23 ∞ 27 6 41 40

D 35 13 27 ∞ 30 16 37

E 51 47 6 30 ∞ 31 7

F 18 43 41 16 31 ∞ 33

G 34 10 40 37 7 33 ∞

Table-1



41

From the Table-1, we have a complete graph of seven vertices, which is shown in Figure-8.

Now, applying the Step 2 of the algorithm, we have a graph as shown in Figure-9.

Now, applying the Step 6 to Step 8 of the algorithm, we have a graph as shown in Figure-10 and this

is the least cost route as A→C→E→G→B→D→F→A with weights equal to 63.

Figure 8 Figure 9 Figure 10

Case 2: The following cost matrix (Table-2) is considered for nine cities (i.e., for 4=m of 12 +mK

and 133912 ⋅==+m )

A B C D E F G H I

A ∞ 31 2 81 59 80 86 29 57

B 31 ∞ 34 21 88 78 76 91 20

C 2 34 ∞ 39 7 93 73 70 97

D 81 21 39 ∞ 41 24 84 69 64

E 59 88 7 41 ∞ 47 13 90 61

F 80 78 93 24 47 ∞ 49 26 95

G 86 76 73 84 13 49 ∞ 52 18

H 29 91 70 69 90 26 52 ∞ 55

I 57 20 97 64 61 95 18 55 ∞

Table-2

From the Table-2, we have a complete graph of nine vertices, which is shown in Figure-11.

Now, applying the Step 9 of the algorithm, we delete 9 greatest weights attached with

edges, 81=AD , 84=DG , 86=AG , 88=BE , 90=EH , 91=BH , 93=CF , 95=FI , 97=CI

forming 3 edge disjoint 3K . Then we have a graph as shown in Figure-12.

Now, applying the Step10 to Step14 of the algorithm, we delete a cycle 9C whose edges are attached

with greatest weights 59=AE , 61=EI , 64=ID , 69=DH , 70=HC , 73=CG , 76=GB ,

78=BF , 80=FA , and then we delete another cycle 9C whose edges are attached with next greatest

weights 31=AB , 34=BC , 39=CD , 41=DE , 47=EF , 49=FG , 52=GH , 55=HI , 57=IA .

Then we have a graph as shown in Figure-13 and this is the least cost route of the traveler as

A→C→E→G→I→B→D→F→H→A with weights equal to 160.



42

Figure 11 Figure 12 Figure 13

Case 3: The following cost matrix (Table-3) is considered for fifteen cities (i.e., for 7=m of 12 +mK

and 531512 ×==+m )

A B C D E F G H I J K L M N O

A ∞ 94 69 122 44 166 143 3 41 149 169 66 128 93 121

B 94 ∞ 96 81 129 50 170 150 39 37 156 175 48 135 80

C 69 96 ∞ 97 70 136 55 176 158 34 31 163 180 54 141

D 122 81 97 ∞ 99 83 123 61 181 147 29 26 146 184 60

E 44 129 70 99 ∞ 101 71 130 45 185 155 23 21 153 190

F 166 50 136 83 101 ∞ 104 86 138 51 167 162 18 17 161

G 143 170 55 123 71 104 ∞ 105 73 125 57 173 144 14 10

H 3 150 176 61 130 86 105 ∞ 109 87 132 63 178 152 6

I 41 39 158 181 45 138 73 109 ∞ 110 76 139 46 183 159

J 149 37 34 147 185 51 125 87 110 ∞ 112 89 127 53 187

K 169 156 31 29 155 167 57 132 76 112 ∞ 115 77 133 59

L 66 175 163 26 23 162 173 63 139 89 115 ∞ 116 90 140

M 128 48 180 146 21 18 144 178 46 127 77 116 ∞ 118 79

N 93 135 54 184 153 17 14 152 183 53 133 90 118 ∞ 119

O 121 80 141 60 190 161 10 6 159 187 59 140 79 119 ∞

Table: 3

From the Table-3, we have a complete graph of nine vertices, which is shown in Figure-14.



43

Figure: 14

Now, applying the Step 22 of the algorithm, we delete 15 edges forming 5 edge disjoint 3K

attached with greatest weights AE=166, FK=167, AK=169, BG=170, GL=173, BL=175, CH=176,

HM=178, CM=180, DI=181, IN=183, DN=184, EJ=185, JO=187, EO=190. Then we have a regular

sub graph of degree 12 as shown in Figure-15.

Figure-15



44

Now, applying the Step 24 of the algorithm, we delete 30 edges forming 6 edge disjoint cycles

5C attached with next greatest weights AG=143, GM=144, MD=146, DJ=147, AJ=149, BH=150,

HN=146, DJ=147, AJ=149, BH=150, HN=152, NE=153, EK=155, BK=156, CI=158, IO=159,

OF=161, FL=162, CL=163, AD=122, DG=123, GJ=125, JM=127, MA=128, BE=129, EH=130,

HK=132, KN=133, NB=135, CF=136, FI=138, IL=139, LO=140, OC=141. Then we have a regular

sub graph of 15K of degree 8 as shown in Figure-16.

Figure-16

Now, applying the Step 26 of the algorithm, we delete 15 edges forming the cycle 15C

attached with next greatest weights AB=94, BC=96, CD=97, DE=99, EF=101, FG=104, GH=105,

HI=109, IJ=110, JK=112, KL=115, LM=116, MN=118, NO=119, OA=121. Then we have a regular

sub graph of 15K of degree 6 as shown in Figure-17.

Figure-17



45

Applying the Step 28 of the algorithm, we delete 30 edges forming 2 edge disjoint cycles 15C

attached with next greatest weights AC=69, CE=70, EG=71, GI=73, IK=76, KM=77, MO=79,

OB=80, BD=81, DF=83, FH=86, HJ=87, JL=89, LN=90, NA=93, AE=44, EI=45, IM=46, MB=48,

BF=50, FJ=51, JN=53, NC=54, CG=55, GK=57, KO=59, OD=60, DH=61, HL=63, LA=66. Then

we have a graph as shown in Figure-18 and this is the least cost route as

A→H→O→G→N→F→M→E→L→D→K→C→J→B→I→A with weights equal to 349.

Figure-18

6. CONCLUSION

Here we have discussed decomposition of complete graphs 12 +mK for 2≥m into circulant

graphs if 12 +m = a prime number, 1,3312 ≥⋅=+ nmn

and sm 312 =+ , 5≥s is a prime. However,

one can study the other situation like 1,5512 ≥⋅=+ nm n or sm 512 =+ , 7≥s and find the circulant

graph which may lead to study the Hamiltonian circuit related with the traveling salesman problems

in near future.

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