Upcoming SlideShare
×

# 50120130406004 2-3

306 views
224 views

Published on

Published in: Technology
0 Likes
Statistics
Notes
• Full Name
Comment goes here.

Are you sure you want to Yes No
• Be the first to comment

• Be the first to like this

Views
Total views
306
On SlideShare
0
From Embeds
0
Number of Embeds
1
Actions
Shares
0
1
0
Likes
0
Embeds 0
No embeds

No notes for slide

### 50120130406004 2-3

1. 1. International Journal of Computer Engineering COMPUTER ENGINEERING & INTERNATIONAL JOURNAL OFand Technology (IJCET), ISSN 0976-6367(Print), ISSN 0976 - 6375(Online), Volume 4, Issue 6, November - December (2013), © IAEME TECHNOLOGY (IJCET) ISSN 0976 – 6367(Print) ISSN 0976 – 6375(Online) Volume 4, Issue 6, November - December (2013), pp. 25-47 © IAEME: www.iaeme.com/ijcet.asp Journal Impact Factor (2013): 6.1302 (Calculated by GISI) www.jifactor.com IJCET ©IAEME DECOMPOSITION OF COMPLETE GRAPHS INTO CIRCULANT GRAPHS AND ITS APPLICATION 1 Jayanta Kr. Choudhury, 2 Anupam Dutta, 3 Bichitra Kalita 1 Swadeshi Academy Junior College, Guwahati-781005 Patidarrang College, Muktapur, Loch, Kamrup, Assam 3 Department of Computer Application, Assam Engineering College, Guwahati-781013 2 ABSTRACT In this paper, the decomposition of complete graphs K 2m+1 for m ≥ 2 into circulant graphs has been discussed. Two theorems have been established for different values of m ≥ 2 relating to 2m + 1 is prime, 2m + 1 = 3 ⋅ 3n for n ≥ 1 and 2 m + 1 = 3s for s ≥ 5 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 K 2m+1 for m ≥ 2 . Keywords: Algorithm, Circulant Traveling Salesman Problem (TSP). Graphs, Complete Graphs, Hamiltonian Graphs, 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 S n in which S has an element of order n , S ⊆ Z n {0} has vertex set Z n and edge set {[x, x + s ] | x ∈ Z n , s ∈ S }. S. El-Zanati, Kyle King and Jeff Mudrock [3] discussed the cyclic decomposition of circulant graphs into almost bipartite graphs. Dalibor Froncek 25
2. 2. International Journal of Computer Engineering and Technology (IJCET), ISSN 0976-6367(Print), ISSN 0976 - 6375(Online), Volume 4, Issue 6, November - December (2013), © IAEME [4] used labeling to show the cyclic decomposition of complete graphs into K m ,n + e : 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 ( m ) that the circulant graph G cd , d is Hamiltonian decomposable for all positive integers c, d and m with c < d where the graph G( N , d ) has vertex set V = {0,1,2,.........., N − 1} , with {v, w} an edge if v − w ≡ ± d (mod N ) for some 0 ≤ i ≤ log d N  − 1 . 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 2 a 3b graphs, each decomposed into 2 a 3b m 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 K 2 n can be decomposed into n − 2 n -suns, a Hamiltonian cycle and a perfect matching, when n is even and for odd cases, the decomposition is n − 1 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 n n n graph K n , n can be decomposed into n -suns when is even. When is odd, K n , n can be 2 2 2 decomposed into (n − 2 ) n -suns and a Hamiltonian circuit [13]. Tay-Woei Shyu [14] investigated 2 the decomposition of complete graph K n cycles Ct ’ s and stars S k ’ s and studied necessary or sufficient conditions for such a decomposition to exist. Dalibor Froncek [15] show that every bipartite graph H which decomposes K k and K n also decomposes K kn . Darryn E. Bryant and Peter Adams [16] discussed that for all v ≥ 3 , the complete graph K v on v vertices can be decomposed into v − 2 edge disjoint cycles whose lengths are 3,3,4,5,KK, v − 1 and for all v ≥ 7 , K v can be decomposed into v − 3 edge disjoint cycles whose lengths are 3,4,KK, v − 4, v − 2, v − 1, v . Fu, Hwang, Jimbo, Mutoh and Shiue [17] considered the problem of decomposing a complete graph into the Cartesian product of two complete graphs K r and K c 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 K 2m+1 into m edge disjoint circulant graphs C2 m+1 ( j ) for m ≥ 2 where j is the jump, 1 ≤ j ≤ m . Thereafter, we present a theorem for the decomposition of K 2m+1 into edge disjoint circulant graphs whenever 2m + 1 is a prime, 2m + 1 = 3 ⋅ 3n , n ≥ 1 and 2 m + 1 = 3s , s ≥ 5 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. 26
3. 3. International Journal of Computer Engineering and Technology (IJCET), ISSN 0976-6367(Print), ISSN 0976 - 6375(Online), Volume 4, Issue 6, November - December (2013), © IAEME 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 n × n matrix C of the form  c0   cn −1 C = M   c2  c  1 c1 L cn−2 c0 c1 c n−1 c0 O c2 L c n−1 O O c n−1   cn−2  M   c1  c0   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 c n −i = ci for i = 1,2,......., n − 1 . Further, C is an adjacency matrix if c0 = 0 and cn −1 = ci ∈ {0, 1}. 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 C n , the complete graph K n , and the complete bipartite graph K n , n . Circulant Graph 2.3 [27] : For a given positive integer, let n1 , n2 ,KK, nk be a sequence of integers where ( p + 1) . 0 < n1 < n2 < KK < nk < 2 Then the circulant graph C p (n1 , n 2 , KK , n k ) is the graph on p nodes v1 , v 2 , KK , v p with vertex vi adjacent to each vertex v i ± n j ( mod p ) . The values ni are called jump sizes. Definition 2.4[28]: A graph G graphs H 1 , H 2 , H 3 ,KK, H i , 1 ≤ i ≤ k , is said having to be no decomposable into the isolated vertices sub and {E(H1 ), E(H 2 ), E (H 3 ),KK, E (H k )} is a partition of E (G ) . 3. THEOREMS 3.1 Theorem: The complete graph K 2m+1 for m ≥ 2 can be decomposed to m edge disjoint circulant graph C 2 m+1 ( j ) where j is the jump and 1 ≤ j ≤ m . 27
4. 4. International Journal of Computer Engineering and Technology (IJCET), ISSN 0976-6367(Print), ISSN 0976 - 6375(Online), Volume 4, Issue 6, November - December (2013), © IAEME Proof: Consider the complete graph K 5 = K 2×2+1 . Then vertex set of K 5 is V = {v1 , v2 , v3 , v4 , v5 } and n(n − 1) 5 × 4 = = 10 edges. Let us partitioned the edges of K 5 into 2 distinct sets containing it has 2 2 5 edges each as E1 = {v1v2 , v 2 v3 , v3 v4 , v4 v5 , v5 v1 } and E2 = {v1v3 , v2 v4 , v3 v5 , v 4 v1 , v5 v2 } respectively. Adjoining edges of E1 and E 2 , we can construct 2 edge disjoint circulant graphs C5 (1) and C5 (2) for the jumps j = 1 and j = 2 respectively and their respective permutations can be represented as f1 and f 2 where v f1 =  1 v  2 v2 v3 v3 v4 v4 v5 v5  v  and f 2 =  1  v v1   3 v2 v4 v3 v5 v4 v1 v5   v2   i.e., f1 = (v1v2 v3 v4 v5 ) and f 2 = (v1v3 v5 v 2 v 4 ) . Figure-1 shows two circulant graphs C5 (1) and C5 (2) for the complete graph K 5 . C5 (1) C5 (2) Figure-1 Now, we append the edges to the circulant graphs as vi + j if 1 ≤ i ≤ 4 f j (vi ) =  vi −4 if i = 5 vi + j if 1 ≤ i ≤ 3 and f j (vi ) =  vi −3 if 4 ≤ i ≤ 5 , j =1 , j=2 Similarly, consider the complete graph K 7 = K 2×3+1 whose vertex set is V = {v1 , v 2 , v3 , v4 , v5 , v6 , v7 } n(n − 1) 7 × 6 = = 21 edges. Let us partitioned the edges of K 7 into 3 distinct sets and have 2 2 E1 = {v1v2 , v2 v3 , v3 v4 , v4 v5 , v5 v6 , v6 v7 , v7 v1 } , consisting of edges each, as 7 E2 = {v1v3 , v2 v4 , v3 v5 , v4 v6 , v5 v7 , v6 v1 , v7 v2 } and E3 = {v1v4 , v2 v5 , v3 v6 , v4 v7 , v5 v1 , v6 v2 , v7 v3 }. Adjoining 7 edges each of E1 , E 2 and E3 , we can construct 3 edge disjoint circulant graphs C 7 (1) , C7 (2) and C7 (3) for the jumps j = 1 , j = 2 and j = 3 respectively and their respective permutations are f1 , f 2 and f 3 where 28
5. 5. International Journal of Computer Engineering and Technology (IJCET), ISSN 0976-6367(Print), ISSN 0976 - 6375(Online), Volume 4, Issue 6, November - December (2013), © IAEME  v v 2 v3 v 4 v5 v 6 v 7   v1 f1 =  1 v v v v v v v  , f2 = v   3 4 5 6 7 1   2  3  v v 2 v3 v 4 v5 v6 v7  f3 =  1 v v v v v v v   5 6 7 1 2 3  4 v2 v4 v3 v5 v4 v6 f 2 = (v1v3 v5 v7 v2 v4 v6 ) and circulant graphs C 7 (1) , C7 (2) and C7 (3) . i.e., f1 = (v1v2 v3 v4 v5 v6 v7 ) , v1 v6 v1 v7   and v2   f 3 = (v1v4 v7 v3v6 v2 v5 ) . Figure-2 shows v1 v2 v3 v1 v2 v7 v6 v4 v5 v7 v5 C 7 (1) v2 v7 v3 v6 v4 v5 C7 (2) v7 v6 v3 v4 v5 C7 (3) Figure-2 The edges to the circulant graphs can be appended as vi + j if 1 ≤ i ≤ 6 f j (v i ) =  vi −6 if i = 7 vi+ j if 1 ≤ i ≤ 5 f j (vi ) =  vi−5 if 6 ≤ i ≤ 7 and , j =1 , j=2 vi+ j if 1 ≤ i ≤ 4 f j (vi ) =  vi−4 if 5 ≤ i ≤ 7 , j =3 Similarly, partitioning the edges of K 9 into 4 distinct sets containing 9 edges each as E1 = {v1v2 , v2 v3 , v3 v4 , v4 v5 , v5 v6 , v6 v7 , v7 v8 , v8 v9 , v9 v1 } , E2 = {v1v3 , v2 v4 , v3 v5 , v4 v6 , v5 v7 , v6 v8 , v7 v9 , v8 v1 , v9 v2 } , E3 = {v1v4 , v2 v5 , v3 v6 , v4 v7 , v5 v8 , v6 v9 , v7 v1 , v8 v2 , v9 v3 } and E4 = {v1v5 , v 2 v6 , v3 v7 , v4 v8 , v5 v9 , v6 v1 , v7 v2 , v8 v3 , v9 v 4 } respectively, we can construct 4 edge disjoint circulant graphs C9 (1) , C9 (2) , C9 (3) and C9 (4) for the jumps j = 1 , j = 2 , j = 3 and j = 4 respectively and their corresponding permutations are given by 29
6. 6. International Journal of Computer Engineering and Technology (IJCET), ISSN 0976-6367(Print), ISSN 0976 - 6375(Online), Volume 4, Issue 6, November - December (2013), © IAEME v f1 =  1 v  2 v3 v4 v4 v5 v5 v6 v6 v7 v7 v8 v8 v9 v9   = (v1v 2 v3 v 4 v5 v6 v 7 v8 v9 ) , v1   v f2 =  1 v  3 v2 v4 v3 v5 v4 v6 v5 v7 v6 v8 v7 v9 v8 v1 v9   = (v1v3 v5 v 7 v9 v 2 v 4 v6 v8 ) , v2   v f3 =  1 v  4 v2 v3 v4 v5 v6 v7 v8 v5 v6 v7 v8 v9 v1 v2 v9   = (v1v 4 v 7 )(v 2 v5 v8 )(v3 v6 v9 ) v3   v f4 =  1 v  5 and v2 v3 v2 v3 v4 v5 v6 v7 v8 v6 v7 v8 v9 v1 v2 v3 v9   = (v1v5 v9 v 4 v8 v3 v7 v 2 v6 ) v4   respectively, whose structures are shown in Figure-3. V1 V9 V3 V4 C9 (1) V7 V7 V5 V6 V8 V8 V4 V7 V5 V6 C9 (2) Append the edges to the circulant graphs as vi + j if 1 ≤ i ≤ 8 f j (vi ) =  vi −8 if i = 9 , j =1 vi+ j if 1 ≤ i ≤ 7 f j (vi ) =  vi−7 if 8 ≤ i ≤ 9 , j=2 vi+ j if 1 ≤ i ≤ 6 f j (vi ) =  vi−6 if 7 ≤ i ≤ 9 , j =3 vi+ j if 1 ≤ i ≤ 5 vi−5 if 6 ≤ i ≤ 9 V6 C9 (3) Figure-3 and f j (vi ) =  V2 V9 V9 V3 V8 V4 V2 V2 V3 V5 V1 V1 V1 V2 , j=4 30 V9 V3 V8 V4 V7 V5 V6 C9 (4)
7. 7. International Journal of Computer Engineering and Technology (IJCET), ISSN 0976-6367(Print), ISSN 0976 - 6375(Online), Volume 4, Issue 6, November - December (2013), © IAEME From the above discussion, it is seen that the complete graph K 2m+1 for m ≥ 2 can be decomposed into m edge disjoint circulant graph C 2 m+1 ( j ) . Here j represents the jump of the circulant also depends upon m . If m = 2 , then j = 1, 2 and we have 2 circulant graphs C5 (1) and C5 (2) . Again, if m = 3 , then j = 1, 2, 3 and then K 7 decomposes into 3 edge disjoint circulant graphs C 7 (1) , C7 (2) and C7 (3) and so on. Moreover, in their respective permutations, the images of first element and last element in the first row are respectively v j +1 and v j . Here we excluded the identity permutations as well as permutations which produce the same circulant graphs,  v1 v 2 v3 v 4 v5   v1 v 2 v3 v 4 v5  i.e.   v v v v v  and  v v v v v  represents the same circulant graph C5 (1) .    3 4 5 1 1 2 3 4  2  5 Therefore, as discussed above, we get 2 permutations if m = 2 , 3 permutations if m = 3 etc, i.e., number of permutations are depends upon the value of m ≥ 2 . Finally, we can comment that the complete graph K 2m+1 for m ≥ 2 can be decomposed into m edge disjoint circulant graph C 2 m+1 ( j ) for the jump 1 ≤ j ≤ m and their corresponding m permutations are defined as vi+ j if 1 ≤ i ≤ 2m − ( j −1)  f j (vi ) =  vi+( j−1)−2m if 2m − ( j − 2) ≤ i ≤ 2m + 1  for 1 ≤ j ≤ m and which complete the theorem. Note: Here, complete graphs K 2m+1 for m ≥ 2 has odd number of vertices and therefore they play different role for different values of m ≥ 2 . In the next theorem we discuss only three cases. 3.2 Theorem: Let K 2m+1 be a complete graph for m ≥ 2 . Then following conditions are satisfied (a) If p = 2m + 1 is a prime number, then K p can be decomposed into m edge disjoint circulant graphs C p ( j ) where j is the jump and 1 ≤ j ≤ m . 1 n +1 3 − 1 for n ≥ 1 , then K q can be decomposed into 2 1 (i) 3n edge disjoint circulant graphs C q ( j ) for the jump 1 ≤ j ≤ m but j ≠ 3l for 1 ≤ l ≤ 3 n − 1 2 and 1 (ii) a circulant graph C q (3,6,9, KK ,3l ) for the jump 3l where 1 ≤ l ≤ 3 n − 1 . 2 (c) If r = 2m + 1 = 3s where s ≥ 5 is a prime number, then K r can be decomposed into (i) s − 1 edge disjoint circulant graphs C r ( j ) for the jump 1 ≤ j ≤ m where j ≠ s and 1 j ≠ 3e , 1 ≤ e ≤ (m − 1) 3 1 (ii) a circulant graph C r (3,6,9,KK,3e) for the jump 1 ≤ e ≤ (m − 1) and 3 (iii) a circulant graph C r (s ) for the jump s . (b) If q = 2m + 1 = 3 ⋅ 3n where m = ( ) ( ( 31 ) )
8. 8. International Journal of Computer Engineering and Technology (IJCET), ISSN 0976-6367(Print), ISSN 0976 - 6375(Online), Volume 4, Issue 6, November - December (2013), © IAEME Proof: (a) When p = 2m + 1 = prime, then it follows that the complete graph K 5 , K 7 , K11 , K13 ,……….. for m = 2,3,5,6,KKK exist and the proof is same as discussed in Theorem 3.1. Hence the complete graph K p where p = 2m + 1 is a prime for m ≥ 2 can be decomposed into 1 m = ( p − 1) edge disjoint circulant graphs C p ( j ) for the jump 1 ≤ j ≤ m . 2 In the above discussion, we observed that each edge disjoint circulant graph C p ( j ) for the 1 ( p − 1) is regular of degree two. In other words, they are cycle C p of length p and 2 therefore they represent edge disjoint Hamiltonian circuits. So in this discussion, number of edge disjoint Hamiltonian circuits in the complete graph K p is m if p = 2m + 1 is a prime for m ≥ 2 . jump 1 ≤ j ≤ Proof: (b) Let K 2m+1 be a complete graph for m ≥ 2 . Suppose that q = 2m + 1 = 3 ⋅ 3n such that 1 v m = (3n +1 − 1) for n ≥ 1 . Then V = { , v , v ,......... .., v } is the vertex set of K and it has q(q − 1) 1 2 2 3 q q 2 edges. Let us define f j :V (K q )→ V (K q ) as follows: (i) when 1 ≤ j ≤ 1 (3 n + 1 − 1 ) but j ≠ 3l for 1 ≤ l ≤ 1 (3n − 1) 2 2 ∀n ≥1 vi + j if 1 ≤ i ≤ q − j  f j (v i ) =  vi + j − q if q − j + 1 ≤ i ≤ q  vi +3l if 1 ≤ i ≤ q − 3l f 3l (vi ) =  vi +3l − q if q − 3l + 1 ≤ i ≤ q and (ii) when 1 ≤ l ≤ 1 (3 n − 1) ∀n ≥ 1 , 2 For n = 1 , m = 4 and q = 2 × 4 + 1 = 9 = 3 ⋅ 31 . Then {v1 , v2 , v3 .v4 , v5 , v6 , v7 , v8 , v9 } is the vertex set of K 9 have q(q − 1) = 9 × 8 = 4 × 9 = 36 edges. Since m = 4 , we can partition the edges of K 9 into 4 edge 2 2 disjoint sets, each set containing 9 edges, by the above rule and their permutations are given as follows. v For j = 1 , f1 =  1 v  2 v For j = 2 , f 2 =  1 v  3 v v2 v3 v4 v5 v6 v7 v3 v4 v5 v6 v7 v8 v9   = (v1v 2 v3 v4 v5 v6 v7 v8 v9 ) v1   v8 v9 v2 v3 v4 v5 v6 v7 v4 v5 v6 v7 v8 v9 v1 v v v4 v5 v6 v7 v8 v8 v9 v1 v2 v3 v9   = (v1v3 v5 v7 v9 v 2 v 4 v6 v8 ) v2   v8 For j = 4 , f 4 =  1 2 3 v v v 6 7  5 v9   = (v1v5 v9 v 4 v8 v3 v7 v2 v6 ) v4   Again, for l = 1 , f 3 =  v1 v2 v3 v4 v5 v6 v7 v8 v9  = (v1v4 v7 )(v2 v5 v8 )(v3 v6 v9 )      v4 v5 v6 v7 v8 v9 v1 v2 32 v3 
9. 9. International Journal of Computer Engineering and Technology (IJCET), ISSN 0976-6367(Print), ISSN 0976 - 6375(Online), Volume 4, Issue 6, November - December (2013), © IAEME From these permutations, we get 4 edge disjoint sets E1 = {v1v2 , v2 v3 , v3 v4 , v4 v5 , v5 v6 , v6 v7 , v7 v8 , v8 v9 , v9 v1 } , E2 = {v1v3 , v2 v4 , v3 v5 , v4 v6 , v5 v7 , v6 v8 , v7 v9 , v8 v1 , v9 v2 } , E3 = {v1v4 , v2 v5 , v3 v6 , v4 v7 , v5 v8 , v6 v9 , v7 v1 , v8 v2 , v9 v3 } and E4 = {v1v5 , v2 v6 , v3 v7 , v4 v8 , v5 v9 , v6 v1 , v7 v2 , v8 v3 , v9 v4 } . Adjoining edges of E1 , E 2 and E 4 , we get 31 = 3 edge disjoint circulant graphs C9 (1) , C9 (2) and C9 (4) for the jumps j = 1 , j = 2 and j = 4 respectively [Figure-3]. Also, adjoining the edges of E3 , we get a circulant graph C9 (3) for the jump 3l where l = 1 . Thus K 9 decomposed into 31 edge disjoint circulant graphs C9 ( j ) for 1 ≤ j ≤ 4 , j ≠ 3 and a circulant graph C9 (3l ) for l = 1 [Figure-3]. For n = 2 , m = 13 and q = 2 × 13 + 1 = 27 = 3 ⋅ 32 . Then {v1 , v2 , v3 ,KKKv 27 } is the vertex set of the complete graph K 27 and have q (q − 1) = 27 × 26 = 13 × 27 = 351 edges. Since m = 13 , we can 2 2 partition the edges of K 27 into 13 edge disjoint sets, each set containing 27 edges, by the rule defined above. For j = 1 , f1 = (v1v2v3v4v5v6v7v8v9v10v11v12v13v14v15v16v17v18v19v20v21v22v23v24v25v26v27 ) For j = 2 , f 2 = (v1v3v5v7v9v11v13v15v17v19v21v23v25v27v2v4v6v8v10v12v14v16v18v20v22v24v26 ) For j = 4 , f4 = (v1v5v9v13v17v21v25v2v6v10v14v18v22v26v3v7v11v15v19v23v27v4v8v12v16v20v24 ) For j = 5 , f 5 = (v1v6 v11v16 v 21v 26 v 4 v9 v14 v19 v 24 v 2 v7 v12 v17 v 22 v 27 v5 v10 v15 v 20 v 25 v3 v8 v13 v18 v 23 ) For j = 7 , f 7 = (v1v8 v15 v22 v2 v9 v16 v 23v3 v10 v17 v24 v4 v11v18v 25v5 v12 v19 v26 v6 v13v20 v27 v7 v14 v21 ) For j = 8 , f 8 = (v1v9 v17 v25v6 v14 v22 v3 v11v19 v27 v8 v16v 24v5 v13v21v2 v10 v18v26 v7 v15v23v4 v12 v20 ) For j = 10 , f10 = (v1v11v21v4 v14v24v7 v17v27v10v20v3v13v23v6 v16v26v9 v19v2 v12v22v5 v15v25v8 v18 ) For j = 11 , f11 = (v1v12v23v7 v18v2 v13v24v8 v19v3v14v25v9 v20v4 v15v26v10v21v5 v16v27v11v22v6 v17 ) For j = 13 , f13 = (v1v14 v 27 v13v26 v12 v25 v11v 24 v10 v 23v9 v22 v8 v 21v7 v20 v6 v19 v5 v18 v4 v17 v3 v16 v 2 v15 ) Again, For l = 1 , f 3 = (v1v 4 v 7 v10 v13 v16 v19 v 22 v 25 )(v 2 v5 v8 v11v14 v17 v 20 v 23 v 26 )(v3 v 6 v9 v12 v15 v18 v 21v 24 v 27 ) For l = 2 , f 6 = (v1v7 v13v19v25v4 v10v16v22 )(v2 v8 v14 v20v26v5 v11v17 v23 )(v3v9 v15v21v27 v6 v12v18v24 ) For l = 3 , f 9 = (v1v10v19 )(v2v11v20 )(v3v12v21 )(v4v13v22 )(v5v14v23 )(v6v15v24 )(v7v16v25 )(v8v17v26 )(v9v18v27 ) For l = 4 , f12 = (v1v13v25v10v22v7 v19v4v16 )(v2v14v26v11v23v8v20v5v17 )(v3v15v27v12v24v9v21v6v18 ) 33
10. 10. International Journal of Computer Engineering and Technology (IJCET), ISSN 0976-6367(Print), ISSN 0976 - 6375(Online), Volume 4, Issue 6, November - December (2013), © IAEME From these permutations we can define corresponding edge disjoint sets E1 , E 2 , E 3 , E 4 , E5 , E 6 , E 7 , E8 , E 9 , E10 , E11 , E12 and E13 respectively. Adjoining edges of E1 , E 2 , E 4 , E 5 , E 7 , E8 , E10 , E11 and E13 , we can construct 32 = 9 edge disjoint circulant graphs C27 (1) , C27 (2) , C27 (4) , C27 (5) , C27 (7 ) , C27 (8) . C27 (10) , C27 (11) and C27 (13) for the jumps j = 1 , j = 2 , j = 4 , j = 5 , j = 7 , j = 8 , j = 10 , j = 11 and j = 13 respectively. Also, adjoining edges of E 4 , E 6 , E 9 and E12 , we get a circulant graph C27 (3,6,9,12) for the jump 3l where 1 ≤ l ≤ 4 . Thus the complete graph K 27 can be decomposed into 32 = 9 edge disjoint circulant graph C27 ( j ) where j is the jump such that 1 ≤ j ≤ 13 , j ≠ 3,6,9,12 and a circulant graph C27 (3,6,9,12) . Let n = k so that m = 1 (3 k +1 − 1) and q = 2m + 1 = 3 ⋅ 3k = 3k +1 . Then {v1 , v 2 , v 3 , KKK , v q } is 2 k +1 k +1 the vertex set the complete graph K q = K 3k +1 and it has q(q − 1) = 3 (3 − 1) edges. 2 2 1 k +1 Since m = 3 − 1 , so we can partition edges of K q = K 3k +1 into 1 (3 k +1 − 1) distinct sets, 2 2 ( ) each set containing 3 k +1 edges, given by (i) when 1 ≤ j ≤ 1 (3 k + 1 − 1 ) but j ≠ 3l for 1 ≤ l ≤ 1 (3 k − 1) 2 2 vi + j if 1 ≤ i ≤ q − j  f j (vi ) =  vi + j − q if q − j + 1 ≤ i ≤ q  and (ii) when 1 ≤ l ≤ 1 (3k − 1) 2 v i + 3l if 1 ≤ i ≤ q − 3l f 3l (v i ) =  v i + 3l − q if q − 3l + 1 ≤ i ≤ q Now, we can have edge disjoint sets E1 , E 2 , E3 , E 4 ,……………, E 1 2 f1 , f 2 , f 3 , f 4 ,……………., f 1 corresponding permutations are 2 E1 , E 2 , E 4 , E 5 , E 7 ,……, E 1 (3 2 k +1 −5 ) , E1 (3 2 k +1 −1 ) (3 k +1 −1 ) (3 k +1 −1 ) and their . Adjoining edges of , we can construct 3 k edge disjoint circulant graphs 1 C 3k +1 ( j ) where j is the jump and 1 ≤ j ≤ 1 (3 k + 1 − 1) , j ≠ 3l for 1 ≤ l ≤ (3 k − 1) . Also, adjoining 2 2 edges of E 3 , E 6 , E 9 ,……, E 1 2 jump 1 ≤ l ≤ (3 k +1 −7 ) , E1 2 (3 k +1 −3 ) , we get a circulant graph C 3k +1 (3,6,9, KK ,3l ) for the 1 k 3 −1 . 2 ( ) ( ) Now, adding 2 ⋅ 3k +1 additional vertices to the vertex set and 4 ⋅ 9 k +1 − 3k +1 additional edges to the edge set of the complete graph K 3k +1 . Then {v1 , v 2 , v3 ,K, v3k +1 , v3k + 2 +1 ,K, v3k + 2 −1 , v3k + 2 } will be the vertex set for K 3k + 2 = K 3⋅3k +1 i.e., K q = K 3k + 2 and the complete graph q = 3 k + 2 ⇒ 2m + 1 = 3 k + 2 1 1 ⇒ m = 3k + 2 − 1 = 3(k +1)+1 − 1 2 2 ( ) { }. The complete graph 34 K 3k + 2 has
11. 11. International Journal of Computer Engineering and Technology (IJCET), ISSN 0976-6367(Print), ISSN 0976 - 6375(Online), Volume 4, Issue 6, November - December (2013), © IAEME ( ) { } q(q − 1) 3k +2 3k +2 − 1 3(k +1)+1 3(k +1)+1 − 1 edges. These edges can be partition into = = 2 2 2 1 k +2 (3 − 1) = 1 {3(k +1)+1 − 1} disjoint sets and their respective permutations are given by 2 2 (i) when 1 ≤ j ≤ 1 (3 k +2 − 1) but j ≠ 3l for 1 ≤ l ≤ 1 (3 k +1 − 1) 2 2  vi + j if 1 ≤ i ≤ q − j f j (vi ) =  vi + j − q if q − j + 1 ≤ i ≤ q  v if 1 ≤ i ≤ q − 3l 1 k +1 and (ii) when 1 ≤ l ≤ 3 − 1 , f 3l (vi ) =  i +3l 2 vi +3l − q if q − 3l + 1 ≤ i ≤ q ( Now, we can have ) 1 (k +1)+1 3 − 1 edge disjoint sets E1 , E 2 , E 3 , E 4 , …, E 1 k + 2 and their respective (3 −1) 2 2 { } permutations are f1 , f 2 , f 3 , f 4 ,………, f 1 2 E 7 ,…….., E 1 (3 2 k +2 −7 ) , E1 (3 2 k +2 −5 ) , E1 (3 2 k+2 −1 ) (3 k +2 −1 ) . Adjoining edges of E1 , E 2 , E 4 , E5 , , we can construct 3 k +1 edge disjoint circulant graph C 3k + 2 ( j ) i.e., 1 k +1 3 − 1 . Also, adjoining 2 , we have a circulant graph C 3( k +1 )+1 (3,6,9, KK ,3l ) for the C 3( k +1 )+1 ( j ) where j is the jump and 1 ≤ j ≤ 1 (3 (k +1)+1 − 1) , 2 edges of E3 , E 6 , E 9 ,….., E 1 2 (3 k +2 −9 ) , E1 2 (3 k +2 −3 ) ( j ≠ 3l for 1 ≤ l ≤ ) jump 1 ≤ l ≤ 1 (3 k + 1 − 1 ) . Thus the result is true for n = k + 1 whenever it is true for n = k . Hence by 2 principle of mathematical induction, the result is true for all n ∈ N . In this discussion, we observed that each edge disjoint circulant graph C q ( j ) for the jump 1≤ j ≤ 1 n +1 3 −1 2 ( ), j ≠ 3l for 1≤ l ≤ 1 n 3 −1 2 ( ) are regular of degree two. In other words they are cycle Cq of length q and therefore they represent 3n edge disjoint Hamiltonian circuits n ≥ 1 . Further the circulant graph C q (3 , 6 , 9 , K K , 3 l ) 1 n 3 −1 2 ( ) for the jump 1 ≤ l ≤ 1 (3 n − 1) , as we have observed, can be decomposed into 2 edge disjoint circulant graphs C q (3t ) for the jump 3t , 1 ≤ t ≤ 1 n 3 − 1 for all n ≥ 1 . If the jump 2 ( ) is 3t = 3 n , then 3n K 3 -decomposition 3t = 3n−1 , then 3n C9 -decomposition 3t = 3n−2 , then 3n C27 -decomposition ……………………………………….. ……………………………………….. 3t = 33 , then 3n C 3 -decomposition n−2 3t = 32 , then 3n C 3 -decomposition n −1 1 n 3t = 3 , then 3 C 3 -decomposition Clearly above cycles does not represent Hamiltonian circuits. So in this discussion, number of edge disjoint Hamiltonian circuits in the complete graph K q is n if q = 3⋅ 3n for n ≥ 1 . n 35
12. 12. International Journal of Computer Engineering and Technology (IJCET), ISSN 0976-6367(Print), ISSN 0976 - 6375(Online), Volume 4, Issue 6, November - December (2013), © IAEME (c) Let K 2 m+1 be a complete graph for all m ≥ 2 . Suppose that r = 2m + 1 = 3s where s ≥ 5 is a prime number and m = 3 s − 1 . Then V = {v1 , v2 , v3 ,..........., vr } is the vertex set of the complete graph 2 K r i.e., K 3s and have r (r − 1 ) 2 edges. Let us define f j : V (K s ) → V (K s ) as follows: (i) when 1 ≤ j ≤ 1 (3s − 1) but j ≠ s and j ≠ 3e for 1 ≤ e ≤ 1 (s − 1) 2 2  v i + j if 1 ≤ i ≤ r − j  f j (v i ) =   v i + j − r if r − j + 1 ≤ i ≤ r  (ii) when 1 ≤ e ≤ 1 (s − 1) , f 3 e (v i ) = 2  v i + 3 e if 1 ≤ i ≤ r − 3 e   v i + 3 e − r if r − 3 e + 1 ≤ i ≤ r vi + s if 1 ≤ i ≤ r − s and (iii) f s (v i ) =  vi + s − r if r − s + 1 ≤ i ≤ r For s = 5 , a prime number, we have r = 3 × 5 = 15 and m = 3 × 5 − 1 = 7 . 2 Then {v1 , v2 , v3 ,..........., v14 , v15 } is the vertex set of the complete graph K15 and have r (r − 1) 15 × 14 = = 7 × 15 = 105 edges. Since m = 7 , we can partition the edges of K15 into 7 2 2 disjoint sets, each set containing 15 edges and their corresponding permutations by the above rule are as follows: For j = 1 , f1 = (v1v2 v3 v4 v5 v6 v7 v8 v9 v10 v11v12 v13v14 v15 ) and E1 = {v1v2 , v2 v3 , v3v4 , v4 v5 , v5 v6 , v6 v7 , v7 v8 , v8 v9 , v9 v10 , v10v11 , v11v12 , v12 v13 , v13v14 , v14v15 , v15v1 } For j = 2 , f 2 = (v1v3 v5 v7 v9 v11v13v15v2 v4 v6 v8 v10 v12 v14 ) and E2 = {v1v3 , v2v4 , v3v5 , v4v6 , v5v7 , v6v8 , v7v9 , v8v10 , v9v11, v10v12 , v11v13 , v12v14 , v13v15 , v14v1 , v15v2 } For j = 4 , f 4 = (v1v5 v9 v13v2 v6 v10 v14 v3 v7 v11v15v 4 v8 v12 ) and E4 = {v1v5 , v2 v6 , v3v7 , v4 v8 , v5 v9 , v6 v10 , v7 v11 , v8 v12 , v9 v13 , v10v14 , v11v15 , v12v1 , v13v2 , v14 v3 , v15v4 } For j = 7 , f 7 = (v1v8 v15 v7 v14 v6 v13v5 v12 v 4 v11v3 v10 v2 v9 ) and E7 = {v1v8 , v2 v9 , v3 v10 , v4 v11 , v5 v12 , v6 v13 , v7 v14 , v8 v15 , v9 v1 , v10 v2 , v11v3 , v12 v4 , v13v5 , v14 v6 , v15v7 } For e = 1 , f 3 = (v1v4 v7 v10 v13 )(v2 v5 v8 v11v14 )(v3 v6 v9 v12 v15 ) and E3 = {v1v4 , v2 v5 , v3 v6 , v4 v7 , v5 v8 , v6 v9 , v7 v10 , v8 v11 , v9 v12 , v10 v13 , v11v14 , v12v15 , v13v1 , v14 v2 , v15v3 } For e = 2 , f 6 = (v1v7 v13v4 v10 )(v2 v8 v14 v5 v11 )(v3 v9 v15 v6 v12 ) and E6 = {v1v7 , v2 v8 , v3v9 , v4 v10 , v5 v11 , v6 v12 , v7 v13 , v8 v14 , v9 v15 , v10 v1 , v11v2 , v12 v3 , v13v4 , v14 v5 , v15v6 } For s = 5 , f 5 = (v1v6 v11 )(v2 v7 v12 )(v3 v8 v13 )(v 4 v9 v14 )(v5 v10 v15 ) and E5 = {v1v6 , v2 v7 , v3v8 , v4 v9 , v5 v10 , v6 v11 , v7 v12 , v8 v13 , v9 v14 , v10 v15 , v11v1 , v12 v2 , v13v3 , v14 v4 , v15v5 } 36
13. 13. International Journal of Computer Engineering and Technology (IJCET), ISSN 0976-6367(Print), ISSN 0976 - 6375(Online), Volume 4, Issue 6, November - December (2013), © IAEME Adjoining edges of E1 , E 2 , E 4 and E 7 , we can construct s − 1 = 5 − 1 = 4 edge disjoint circulant graph C15 (1) , C15 (2) , C15 (4) and C15 (7 ) respectively for jumps j = 1 , j = 2 , j = 4 and j = 7 [Figure-5]. Also, adjoining edges of E3 and E 6 , we can construct a circulant graph C15 (3,6) [Figure-6]. Again, adjoining the edges of E 5 , we can construct a circulant graph C15 (5) for the jump s = 5 [Figure-7]. V1 V1 V2 V2 V15 V3 V4 V13 V12 V5 V1 V15 V2 V3 V14 V14 V4 V13 V4 V14 V13 V13 V4 V12 V5 V12 V5 V15 V3 V14 V12 V5 V1 V2 V15 V3 V6 V6 V6 V11 V11 V6 V11 V11 V7 V10 V7 V9 V10 V7 V8 C15 (1) V10 V7 V9 V8 V10 V8 V9 V C15 (2) C15 (4) C15 (7 ) Figure-5 V1 V2 V2 V15 V1 V3 V14 V4 V13 V13 V12 V5 V 11 V6 V1 0 V7 V8 V14 V4 V12 V5 V6 V15 V3 V11 V10 V7 V9 V8 C15 (3,6) Figure-6 V9 C15 (5) Figure-7 Thus the complete graph K15 can be decomposed into s − 1 = 5 − 1 = 4 edge disjoint circulant graphs C15 ( j ) for 1 ≤ j ≤ 7 , j ≠ 3,5,6 and a circulant graph C15 (3,6) and a circulant graph C15 (5) for s = 5 . Similarly, we can show for s = 7,11,13,KK etc. Let s = k , a prime number. Then r = 3k , m = 3 k − 1 2 r (r − 1 ) 3 k (3 k − 1 ) = 2 2 and the complete graph K 3k edges. Since m = 3 k − 1 , so we can partition the edges of K 3k into 1 (3 k − 1 ) 2 2 edge disjoint sets, each set containing 3k edges and their respective permutations are given by (i) when 1 ≤ j ≤ f (ii) when j has 1 (3k − 1) but j ≠ k and j ≠ 3e for 1 ≤ e ≤ 1 (k − 1) 2 2 (v i ) = 1≤ e ≤  v i + j if 1 ≤ i ≤ r − j    v i + j − r if r − j + 1 ≤ i ≤ r  1 (k − 1 ) , 2  v i + 3 e if 1 ≤ i ≤ r − 3 e f 3 e (v i ) =   v i + 3 e − r if r − 3 e + 1 ≤ i ≤ r and (iii) f k (v i ) =  v i + k if 1 ≤ i ≤ r − k   v i + k − r if r − k + 1 ≤ i ≤ r 37
14. 14. International Journal of Computer Engineering and Technology (IJCET), ISSN 0976-6367(Print), ISSN 0976 - 6375(Online), Volume 4, Issue 6, November - December (2013), © IAEME Now, we can have edge disjoint sets E1 , E 2 , E3 , E 4 ,……, E 1 2 are f1 , f 2 , f 3 , f 4 ,………, f 1 2 ( 3 k −1 ) and their respective permutations (3 k − 1 ) . Adjoining edges of E j ’s where 1≤ j ≤ 1 (3 k − 1 ) ; 2 j ≠ k and j ≠ 3e for 1 ≤ e ≤ 1 (k − 1 ) , we can construct s − 1 = k − 1 edge disjoint circulant graphs C3k ( j ) 2 where j is the jump. Again, adjoining the edges of E 3e ’s, 1≤ e ≤ 1 (k − 1 ) 2 we can construct a circulant graph C3k (3,6,9,KK,3e ) for the jump 3e . Also, adjoining the edges of E k , we can construct a circulant graph C3k (k ) for the jump k . In this discussion we observed that each edge disjoint circulant graphs C r ( j ) for the jump 1≤ j ≤ 1 (3 s − 1 ) , j ≠ s and 2 j ≠ 3e for 1 ≤ e ≤ 1 (s − 1) where s ≥ 5 is a prime, are regular of degree 2 two. In other words they are cycles C r of length r and therefore they represent edge disjoint Hamiltonian circuits. Again the circulant graph C r (3,6,9, KK,3e ) for the jump 3e where 1 1 ≤ e ≤ (s − 1 ) can be decomposed into (m − s ) edge disjoint circulant graphs 3t where 2 1 1 ≤ t ≤ (s − 1 ) 2 and these circulants are regular of degree two each and they can be further decomposed into (m − 1) cycles C s of length s < r and so they do not represent Hamiltonian circuits. Also, the circulant graph C r (s ) for the jump s is regular of degree two and from which we can have s K 3 -decompositions and they do not represent a Hamiltonian circuit. So in this discussion, number of edge disjoint Hamiltonian circuits in the complete graph K r is s − 1 if r = 3s , s ≥ 5 is a prime. 4. ALGORITHM Input: Let K 2 m+1 for m ≥ 2 be a complete graph. Output: Find least cost route. Case 1: When p = 2m + 1 is a prime number and (2m + 1) consecutive greatest weights are incident at different vertices of the complete graph K 2 m+1 for m ≥ 2 along edges of a cycle C 2 m+1 , next (2m + 1) greatest weights incident at different vertices along edges of a cycle C 2 m+1 other than the previous one, next (2m + 1) greatest weights incident at different vertices along edges of a cycle C 2 m+1 other than the previous two cycles and so on. Step 1: Study the graph K 2m+1 for m ≥ 2 . Step 2: Delete (2m + 1) number of consecutive greatest weights incident at different vertices along the edges of a cycle C 2 m+1 of the complete graph K 2m+1 for m ≥ 2 . Step 3: Observed that the graph obtained in Step 2 is a regular sub graph of the complete graph K 2m+1 of degree 2 . Otherwise go to Step 6. Step 4: Observed that the graph obtained in Step 3 is a cycle C 2 m+1 for m ≥ 2 attached with minimum weights and this is the required Hamiltonian circuit i.e., least cost route. Step 5: Stop 38
15. 15. International Journal of Computer Engineering and Technology (IJCET), ISSN 0976-6367(Print), ISSN 0976 - 6375(Online), Volume 4, Issue 6, November - December (2013), © IAEME Step 6: From the graph obtained in Step 2, delete another (2m + 1) number of edges attached with consecutive greatest weights other than (2m + 1) 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 q = 2m + 1 = 3 ⋅ 3n for n ≥ 1 and 3 n +1 number of consecutive greatest weights incident at different vertices along edges of 3n edge disjoint K 3 , next 3 n + 2 number of consecutive greatest weights attached with edges of 3n edge disjoint cycles C 32 , next 3 n +3 number of consecutive greatest weights attached with edges of 3n edge disjoint cycles C 33 and so on the remaining 32 n +1 number of consecutive greatest weights are attached with edges of 3n circuits C 3n +1 . Step 9: Delete 3n +1 number of consecutive greatest weights incident at different vertices along edges of 3n edge disjoint K 3 . Step 10: Observed that the graph obtained in Step 9 is a regular sub graph of the complete graph K 3⋅3n of degree 2 ⋅ 3n for n ≥ 1 . Step 11: From the graph obtained in Step 11, delete another 3 n +1 number of edges forming the circuit C 3n +1 attached with consecutive greatest weights other than 3n +1 consecutive greatest weights ( ) already deleted in Step 9, successively 3n − 1 times. Step 12: Observed that the graph obtained in Step 11 is a regular sub graph of the complete graph K 3⋅3n of degree 2 for n ≥ 1 . Otherwise go to Step 15. Step 13: Observed that graph obtained in Step 12 is a circuit C 3n +1 attached with minimum weights, which is the required least cost route. Step 14: Go to Step 5. Step 15: Observed that the graph obtained in Step 10 is a regular sub graph of the complete graph K 3⋅3n of degree 3 3n+1 − 1 for n ≥ 1 . ( ) Step 16: From the graph obtained in Step 15, delete another 3 n + 2 number of consecutive greatest weights other than the greatest weights already deleted in Step 11, incident at different vertices forming 3n edge disjoint cycles C 32 for n ≥ 1 . Step 17: Go to Step 10 to Step 14. Otherwise go to Step 18. Step 18: From the graph obtained in Step 17, delete another 3 n +3 number of consecutive greatest weights other than the greatest weights already deleted in Step 16, incident at different vertices forming 3n edge disjoint cycles C 33 for n ≥ 1 . Step 19: Go to Step 10 to Step 14. Otherwise go to Step 20. Step 20: From the graph obtained in Step 18, delete another 3 n + 4 number of consecutive greatest weights other than the greatest weights already deleted in Step 18, incident at different vertices forming 3n edge disjoint cycles C 34 for n ≥ 1 and continue this process till the graph becomes regular of degree 2 ⋅ 3n for n ≥ 1 . Step 21: Go to Step 10 to Step 14. 39
16. 16. International Journal of Computer Engineering and Technology (IJCET), ISSN 0976-6367(Print), ISSN 0976 - 6375(Online), Volume 4, Issue 6, November - December (2013), © IAEME Case 3: When r = 2m + 1 = 3s , s ≥ 5 is a prime and 3s number of consecutive greatest weights incident at different vertices forming s edge disjoint K 3 , next 3s(m − s ) number of consecutive greatest weights incident at different vertices forming 3(m − s ) edge disjoint cycles C s and the remaining 3s (s − 1) number of consecutive greatest weights incident at different vertices forming (m − s ) edge disjoint cycles C3s . Step 22: Delete 3s number of consecutive greatest weights incident at different vertices forming s edge disjoint K 3 for s ≥ 5 . Step 23: Observed that the graph obtained in Step 22 is a regular sub graph of the complete graph K 3s of degree 3(s − 1) for s ≥ 5 . Step 24: From the graph obtained in Step 23, delete another 3s(m − s ) number of edges forming 3(m − s ) edge disjoint circuits C s attached with greatest consecutive weights, other than 3s number of consecutive greatest weights already deleted in Step 22. Step 25: Observed that the graph obtained in Step 24 is a regular sub graph of the complete graph K 3s of degree 2(s − 1) having 3s (s − 1) number of remaining edges for s ≥ 5 . Step 26: From the graph obtained in Step 25, delete another 3s number of edges forming the cycle C3s attached with consecutive greatest weights which are different from greatest weights already deleted in Step 24. Step 27: Observed that the graph obtained in Step 26 is a regular sub graph of the complete graph K 3s of degree 2(s − 2) for s ≥ 5 . Step 28: Repeat Step 26 on the graph obtained in Step 23 for (s − 3) times where s ≥ 5 . Step 29: Observed that the graph obtained in Step 28 is a cycle C3s 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 m = 3 of K 2m+1 and 2 m + 1 = 7 , 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 40
17. 17. International Journal of Computer Engineering and Technology (IJCET), ISSN 0976-6367(Print), ISSN 0976 - 6375(Online), Volume 4, Issue 6, November - December (2013), © IAEME 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 m = 4 of K 2m+1 and 2m + 1 = 9 = 3 ⋅ 31 ) 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 Table-2 95 18 55 ∞ 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, AD = 81 , DG = 84 , AG = 86 , BE = 88 , EH = 90 , BH = 91 , CF = 93 , FI = 95 , CI = 97 forming 3 edge disjoint K 3 . Then we have a graph as shown in Figure-12. Now, applying the Step10 to Step14 of the algorithm, we delete a cycle C9 whose edges are attached with greatest weights AE = 59 , EI = 61 , ID = 64 , DH = 69 , HC = 70 , CG = 73 , GB = 76 , BF = 78 , FA = 80 , and then we delete another cycle C9 whose edges are attached with next greatest weights AB = 31 , BC = 34 , CD = 39 , DE = 41 , EF = 47 , FG = 49 , GH = 52 , HI = 55 , IA = 57 . 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. 41
18. 18. International Journal of Computer Engineering and Technology (IJCET), ISSN 0976-6367(Print), ISSN 0976 - 6375(Online), Volume 4, Issue 6, November - December (2013), © IAEME Figure 11 Figure 12 Figure 13 Case 3: The following cost matrix (Table-3) is considered for fifteen cities (i.e., for m = 7 of K 2m+1 and 2m + 1 = 15 = 3 × 5 ) A B C D E F G H I A ∞ 94 69 122 44 166 143 B 94 ∞ 96 81 129 50 170 150 C 69 96 ∞ 97 70 136 D 122 81 97 ∞ 99 E 44 129 70 99 F 166 50 136 G 143 170 L M N O 3 41 149 169 66 128 93 121 39 37 156 175 48 135 80 55 176 158 34 31 163 180 54 141 83 123 61 181 147 29 26 146 184 60 ∞ 101 71 130 45 185 155 23 21 153 190 83 101 ∞ 104 86 138 51 167 162 18 17 161 55 123 71 104 ∞ 105 73 125 57 173 144 14 10 61 130 86 105 ∞ 109 87 132 63 178 152 6 45 138 73 109 ∞ 110 76 139 46 183 159 51 125 87 110 ∞ 112 89 127 53 187 57 132 76 112 ∞ 115 77 133 59 63 139 89 115 ∞ 116 90 140 46 127 77 116 ∞ 118 79 152 183 53 133 90 118 ∞ 119 159 187 59 140 79 119 ∞ H 3 150 176 I 41 39 158 181 J 149 37 34 147 185 K 169 156 31 29 155 167 L 66 175 163 26 23 162 173 21 18 144 178 17 14 190 161 10 M 128 48 180 146 N 93 135 54 184 153 O 121 80 141 60 6 J K Table: 3 From the Table-3, we have a complete graph of nine vertices, which is shown in Figure-14. 42
19. 19. International Journal of Computer Engineering and Technology (IJCET), ISSN 0976-6367(Print), ISSN 0976 - 6375(Online), Volume 4, Issue 6, November - December (2013), © IAEME Figure: 14 Now, applying the Step 22 of the algorithm, we delete 15 edges forming 5 edge disjoint K 3 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 43
20. 20. International Journal of Computer Engineering and Technology (IJCET), ISSN 0976-6367(Print), ISSN 0976 - 6375(Online), Volume 4, Issue 6, November - December (2013), © IAEME Now, applying the Step 24 of the algorithm, we delete 30 edges forming 6 edge disjoint cycles C5 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 K15 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 C15 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 K15 of degree 6 as shown in Figure-17. Figure-17 44
21. 21. International Journal of Computer Engineering and Technology (IJCET), ISSN 0976-6367(Print), ISSN 0976 - 6375(Online), Volume 4, Issue 6, November - December (2013), © IAEME Applying the Step 28 of the algorithm, we delete 30 edges forming 2 edge disjoint cycles C15 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 K 2m+1 for m ≥ 2 into circulant graphs if 2m + 1 = a prime number, 2m + 1 = 3 ⋅ 3n , n ≥ 1 and 2 m + 1 = 3s , s ≥ 5 is a prime. However, one can study the other situation like 2m + 1 = 5 ⋅ 5 n , n ≥ 1 or 2 m + 1 = 5 s , s ≥ 7 and find the circulant graph which may lead to study the Hamiltonian circuit related with the traveling salesman problems in near future. REFERENCES [1] [2] Zbigniew R. Bogdanowicz, Decomposition of circulants into antidirected Hamiltonian cycles, Applied Mathematical Sciences, Vol. 6, 2012, no.119, 5935-5938. Matthew Dean, On Hamiltonian cycle decomposition of 6-regular circulant graphs, Graphs and Combinatorics, 22 (2006), 331-340. 45
22. 22. International Journal of Computer Engineering and Technology (IJCET), ISSN 0976-6367(Print), ISSN 0976 - 6375(Online), Volume 4, Issue 6, November - December (2013), © IAEME [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] Saad El-Zanati, Kyle King and Jeff mudrock, On the cyclic decomposition of circulant graphs into almost-bipartite graphs, Australian Journal of Combinatorics, Vol. 49(2011), 61-76. Dalibor Froncek, Cyclic decomposition of complete graphs into K m , n + e : the missing case, Congressus Numerantium, 198 (2009), 111-118. Daniel K. Biss, Hamiltonian decomposition of recursive circulant graphs, Discrete Mathematics, 214(2000), 89-99. C. Micheneau, Disjoint Hamiltonian cycles in recursive circulant graphs, Inform Process Lett. 61 (1997) 259-264. J. C. Bermond, O. Favaron and M. Maheo, Hamiltonian decomposition of cayley graphs of degree 4, Journal of combinatorial theory, series B 46, 142-153 (1989). Marsha F. Foregger, Hamiltonian decompositions of product of cycles, Discrete Mathematics, 24(1978) 251-260. A. Kotzig, Every Cartesian product of two circuits is decomposable into Hamiltonian circuits, manuscript, Centre de Recherché Mathmatique, Montreal (1973). Dave Morris, Hamiltonian cycles in circulant graphs and digraphs, http:// people.uleth.ca/~dave.morris I. Gerace and F. Greco, The traveling salesman problem in symmetric circulant matrices with two stripes, Math. Structures in Comp. Science, 18 (2008), 1-11. I. Gerace and F. Greco, The symmetric circulant traveling salesman problem, “Traveling Salesman Problem”, Book edited by F. Greco, 2008, 181-202. R. Anitha and R. S. Lekshmi, N-Sun Decomposition of Complete Graphs and Complete Bipartite Graphs, World Academy of Science, Engineering and Technology 3(2007) 262-266. Tay-Woei Shyu, Decomposition of Complete Graphs into Cycles and Stars, Graphs and Combinatorics, Vol. 29, Issue 2, 301-313(2011). Dalibor Froncek, Decomposition of Complete Graphs into Small Graphs, Opuscula Mathematica, Vol. 30, No. 3, (2010), 277-280. Darryn E. Bryant and Peter Adams, Decomposing the Complete Graph into Cycles of Many Lenghts, Graph and Combinatoeics, Vol. 11, Issue 2, 97-102(1995). H. L. Fu, F. K. Hwang, M. Jimbo, Y. Mutoh and C. L. Shiue, Decomposition of Complete Graphs into K r × K c ’ s, Journal of statistical planning and inference, 119 (2004), 225-236. Vinh Kha Nguyen, “Family of circulant graphs and its expander properties”, A Thesis, SanJose′ State University, 2010. B. Kalita, “Application of three edges extension of planar (TEEP) graph for the Solution of traveling salesman problem” Proc. 48th ISTAM conference, 2003. B. Kalita, “Non-isomorphic Hamiltonian Sub-Graphs of complete graph and their application” Proc. Indian Sc. Congress, Ahmedabad, 2003. B. Kalita, “Sub Graph of Complete Graph” Proceeding, International Conference on Foundation of computer Science. Lasvegas, U.S.A. p.p.71-77, 2006. B. Kalita, “SCN with DCS tree with cover and Hamiltonian circuit” Proc International workshop on Telemetric 1 - 5th June,1995 p.p. 163 -168, 1995. J. K. Choudhury, B. Kalita, “An Algorithm for TSP” Bulletin of Pure and Applied Sciences, Vol 30 E (Math &Stat.) Issue (No.1) 2011: P. 111-118. J. K.Choudhury, A. Dutta, B. Kalita, “Graph Factorization and Its Application”, International Journals of Multidisciplinary Research Academy, Vol 2, Issue-3 (March 2012) P208-220. A. Dutta, B. Kalita, H. K. Baruah, “Regular sub-graph of complete graph” International Journal of Applied Engineering Research. Vol.5. No-8, (2010) p.p. 1315-1323. 46
23. 23. International Journal of Computer Engineering and Technology (IJCET), ISSN 0976-6367(Print), ISSN 0976 - 6375(Online), Volume 4, Issue 6, November - December (2013), © IAEME [26] A. Dutta, B.Kalita, H.K. Baruah, “Regular Planar Sub-Graphs of Complete Graph and Their Application” .IJAER, Vol-5 no-3, (2010), p.p. 377-386. [27] Fred Buckley, Frank Harary, “Distance in graphs”, Addison Wesley, 1990. [28] G. Chartrand, P. Zhang, “Intoductition to graph theory” Tata McGraw-Hill Edition, 2006. [29] F. Harary, Graph Theory” Narosa Publishing House, 1993. [30] Gulwatanpreet Singh, Surbhi Gupta and Baldeep Singh, “Aco Based Solution for TSP Model for Evaluation of Software Test Suite”, International Journal of Computer Engineering & Technology (IJCET), Volume 3, Issue 3, 2012, pp. 75 - 82, ISSN Print: 0976 – 6367, ISSN Online: 0976 – 6375. [31] Kulbhushan Verma, Manpreet Kaur and Palvee, “Comparative Analysis of Various Types of Genetic Algorithms to Resolve TSP”, International Journal of Electronics and Communication Engineering & Technology (IJECET), Volume 4, Issue 5, 2013, pp. 111 - 116, ISSN Print: 0976- 6464, ISSN Online: 0976 –6472. [32] M.Siva Parvathi and B.Maheswari, “Minimal Dominating Functions of Corona Product Graph of a Cycle with a Complete Graph”, International Journal of Computer Engineering & Technology (IJCET), Volume 4, Issue 4, 2013, pp. 248 - 256, ISSN Print: 0976 – 6367, ISSN Online: 0976 – 6375. 47