International Journal of Computational Science and Mathematics. ISSN 0974-3189 Volume 4, Number 3 (2012), pp. 225-232 © International Research Publication House http://www.irphouse.com

An Algorithm on the Decomposition of Some Complete Tripartite Graphs

R. Franklin Richard and N. Gnana Dhas

Research Scholar, Department of Mathematics Nesamony Memorial Christian College, Marthandam, 629165, Tamil Nadu, India

2HOD, Dept. of Mathematics, Narayanaguru College of Engineering, Tamil Nadu, India

E-mail: franklinsheela@yahoo.co.in n.gnanadhas@gmail.com

Abstract

Let G= (V, E) be a finite connected simple graph and {Gi/i=1,2,n} be a collection of edge-disjoint sub graphs of G such that E(G)=E(G1)UE(G2)U.UE(Gn), then the collection {Gi} is called a decomposition of G. if each Gi is connected and |E(Gi)|=i for each i=1.2,,n, then it is called a continuous monotonic decomposition of G. In this paper we develop an algorithm to compute the necessary and sufficient conditions for for K1,3,m K2,3,m , K2,5,m and K3,5,m and the number of edges and the number of disjoint sets. Keywords: Graph Decomposition, Complete Tripartite graph, Continuous monotonic decomposition, Triangular numbers. AMS Subject Classification: Give at Least two AMS subject codes relating to the broad areas of your paper.

Introduction An undirected simple graph with the property that there is a path between every pair of vertices is known as a connected graph. The degree of a vertex u of any graph is the number of edges incident with u and is denoted by d(u) and the distance between the two vertices u and v of G is the length of the shortest u-v path in G and is denoted by d(u,v). A graph G called a n-regular if deg(v)=n ∀ v�V(G). A complete graph with vertices nN, denoted by Kn , is a connected simple graph with every vertex is connected with every other vertex by an edge. A graph with n vertices v1,v2,…..vn , where n=3, and edges {v1,v2}, {v2,v3},…..{vn-1,vn},{vn,v1} is known as a cycle, Cn.

226 R.Franklin Richard and N.Gnana Dhas

For graph terminology we refer to Harary1

Graph Decomposition Let G=(V,E) be a connected simple graph of order p and size q. If A decomposition (G1,G2,…..Gn) of G is said to be a continuous monotonic decomposition (CMD) if each Gi is connected and |E(Gi)|=i i�N. Alavi2, introduced Ascending Sub graph Decomposition(ASD) as a decomposition of G into subgraphs Gi (not necessarily connected) and is isomorphic to a proper subgraph Gi+1.Gnanadhas and Paulraj Joseph introduced a new concept known as continuous monotonic decomposition of Graphs3. A decomposition, {G1,G2,Gn} n�N is said to be a Continuous Monotonic Decomposition (CMD) if each Gi is connected and |E(Gi)|=i v� i N. If G admits a CMD, {G3,G4,Gn} v� n N, where each Gi is a cycle of length i in G., then we say that G admits Continuous Monotonic Cycle Decomposition (CMCD).4

Figure 1: Example of Graph Decomposition

Continuous Monotonic Decomposition of Complete Tripartite Graphs K2,5,m and K3,5,m Continuous Monotonic Decomposition of a wide variety of graphs had been studied by Gnanadhas and Paulraj Joseph and Navaneetha Krishnan and Nagarajan5 If a graph G admits a CMD {G1,G2,Gn} nN if and only if q=(n+1)C2 The following two results6, Joseph Varghese and A. Antonysamy, are about particular classes of complete tripartite graphs which accept CMD.

An Algorithm on the Decomposition of Some Complete Tripartite Graphs 227

Figure 2: Graph Decomposition of K1,3,13

Figure 3: Graph Decomposition of K2,5,8 Theorem 1: Let G be a connected simple graph of order p and size q. Then G admits a CMD H1,H2,.Hn if and only if q=(n+1)C2.

3

Theorem 2: A complete tripartite graph K1,3,m accepts CMD of H1,H2,H4n+1 if and only if m=(4n2+3n-1)/2 when n is odd and CMD of H1,H2,H4n+2 if and only if m=(4n2+5n)/2 when n is even nN. 6 Theorem 3: A complete tripartite graph K2,3,m accepts CMD of H1,H2,H(5n+7)/2 if and only if m=(5n2+16n+3)/8 when n is odd and CMD of H1, H2 , H(5n+6)/2 if and only if m=(5n2+14n)/8 when n is even nN. 6

228 R.Franklin Richard and N.Gnana Dhas

Theorem 4: A complete tripartite graph K2,5,m accepts CMD of G1,G2,.G7n+2 and G1,G2,G7n+4 if and only if m=(7n2+5n-2)/2 and m=(7n2+9n)/2 respectively nN. 6 Theorem 5: A complete tripartite graph K3,5,m accepts CMD of G1,G2,…..G(16n-6) if and only if m=16n2-11n and CMD of G1, G2 ,….. G16n+5 if and only if m=16n2+11n nN. 6 Proof of Theorem 4: Assume that a complete tripartite graph K2,5,m accepts CMD of G1,G2,…..G7n+2 and G1,G2,…..G7n+4, nN. We have q(K2,5,m ) = [m(2+5)+5(m+2)+2(m+5)]/2 m nN. Case 1: when 7n+2 decompositions K2,5,m accepts CMD of G1,G2,…..G7n+2 iff q(K2,5,m ) = (7n+2)(7n+3)/2 , i.e., 7m+10=(7n+2)(7n+3)/2. i.e., 2m=(7n2+5n-2) i.e., m= (7n2+5n-2)/2 . The values of m are 5, 18, 38, 65, 99 Case 2: when 7n+4 decompositions. K2,5,m accepts CMD of G1,G2,…..G7n+4 iff q(K2,5,m ) = (7n+4)(7n+5)/2 , i.e=(7n+4)(7n+5)/2 =10+7m i.e., m= (7n2+9n)/2 for nN The values of m are 8,23,45,74,110, Hence a complete tripartite graph K2,5,m accepts CMD of G1,G2,. G7n+2 if and only if m=(7n2+5n-2)/2 and CMD of G1,G2,…..G7n+4 if and only if m=(4n2+5n)/2 nN. Conversely Suppose that K2,5,m is a complete tripartite graph with m=(7n2+5n-2)/2 and m=(7n2+9n)/2 nN. We know that q(K2,5,m ) = 10+7m Case 1: when m=(7n2+5n-2)/2 q(K2,5,m ) = 10+7m = 10+7(7n2+5n-2)/2 =(6+49n2+35n)/2 =(7n+2)(7n+3)/2 Which is of the form k(k+1)/2 nN This implies that K2,5,m is a connected simple graph, can be decomposed into G1,G2,Gk nN. i.e., K2,5,m can be decomposed onto G1,G2,.G7n+2 for nN .

An Algorithm on the Decomposition of Some Complete Tripartite Graphs 229

Case 2: when m=(7n2+9n)/2 q(K2,5,m ) = 10+7m = 10+7(7n2+9n)/2 =(20+49n2+63n)/2 =(7n+4)(7n+5)/2 Which is of the form k(k+1)/2 nN This implies that K2,5,m is a connected simple graph, can be decomposed into G1,G2,.Gk nN. i.e., K2,5,m can be decomposed onto H1,H2,.H7n+4 for nN 4. Algorithm 1. K a,b,m Step 1: Read the values of a,b Step 2: if (b==3) then Step a: Initialize n,m,x,y Step 2: Read the value of n If (a==1) then Step 3: for i= 1 to n do if (i%2==0) then i is even

(i)Calculate m= (4n2+5n)/2 (ii) Calculate x=4n+2

Step 4: Else (i) Calculate m= (4n2+3n-1)/2 (ii) Calculate x=4n+1

Step 5: Calculate y=4m+3 Step 6: Print m, y and x Step 7: Goto step 3 until i>n Else if (a==2) then Step 3: for i= 1 to n do if (i%2==0) then i is even

Calculate m= (5n2+14n)/8 x=(5n+6)/2

Step 4: Else (i) Calculate m= (5n2+16n+3)/8 (ii) Calculate x=(5n+7)/2

Step 5: Calculate y=4m+3 Step 6: Print m, y and x Step 7: Go to step 3 until i>n

230 R.Franklin Richard and N.Gnana Dhas

Step: if(b==5) then Step 2: Initialize n,m,x,y Step 2: Read the value of n If (a==2) Calculate m= (7n2+5n-2)/2 (iii) Calculate x=7n+2 (iv) Calculate y=7m+10 (v) (vi) Print m, y and x

Step 3: for i= 1 to n do (iv)Print m, y and x

Step 5: Goto step 3 until i>n Else if(a==3) Step 3: for i= 1 to n do (i)Calculate m= (16n2+11n) (vii)Calculate x=16n+5 (vii)Calculate y=8m+15 (ix)Print m, y and x

Step 4 (i)Calculate m= (16n2-11n) (ii)Calculate x=(16n-6)

Step 5: Calculate y=8m+15 Step 6: Print m, y and x Step 7: Goto step 3 until i>n Step 8: Stop. (1): Output: (K 1,3,m) Enter the value of n 10 a=1,b=3

Sl.No m Q CMD 1 3 15 H1…………H5 2 13 55 H1…………H10 3 22 91 H1…………H13 4 42 171 H1…………H18

An Algorithm on the Decomposition of Some Complete Tripartite Graphs 231

5 57 231 H1…………H21 6 87 351 H1…………H26 7 108 435 H1…………H29 8 148 595 H1…………H34 9 175 703 H1…………H37 10 225 903 H1…………H42

(2): Output: (K 2,3,m) Enter the value of n 10 a=2,b=3

Sl.No M Q CMD 1 3 21 H1…………H6 2 6 36 H1…………H8 3 12 66 H1…………H11 4 17 91 H1…………H13 5 26 136 H1…………H16 6 33 171 H1…………H18 7 45 231 H1…………H21 8 54 276 H1…………H23 9 69 351 H1…………H26 10 80 406 H1…………H28

(3): Output: K2,5,m Enter the value of n 10 a=2,b=5

Sl.No M q CMD 1 5 45 H1…………H9 2 8 66 H1…………H11 3 18 136 H1…………H16 4 23 171 H1…………H18 5 38 276 H1…………H23 6 45 325 H1…………H25 7 65 465 H1…………H30 8 74 528 H1…………H32 9 99 703 H1…………H37 10 110 780 H1…………H39

232 R.Franklin Richard and N.Gnana Dhas

(4): Output: ( K3,5,m ) Enter the value of n 10 a=3, b=5

Sl.No M q CMD 1 5 55 H1…………H10 2 27 231 H1…………H21 3 42 351 H1…………H26 4 86 703 H1…………H37 5 111 903 H1…………H42 6 177 1431 H1…………H53 7 212 1711 H1…………H58 8 300 2415 H1…………H69 9 345 2775 H1…………H74 10 455 3655 H1…………H85

References

[1] F.Harary, Graph Theory, Addision-Wesley Publishing House, USA, 1969 [2] Y.Alavi, “The Ascending Subgraph Decomposition Problem” Congress

Numerantium, 1987, Vol. 58, p.7-14 [3] N.Gnanadhas and J. Paulraj Joseph, “Continuous Monotonic Decomposition of

Graphs”, International Journal of Management and Systems, Vol.16, Sep-Dec, 2000, pp.333-344

[4] N.Gnanadhas and J. Paulraj Joseph, “Continuous Monotonic Decomposition of Cycles”, International Journal of Management and Systems, Vol.19, Jan-Apr, 2003, pp.65-76

[5] A.Nagarajan and S.Navaneetha Krishnan, “Continuous Monotonic Decomposition of some special class of Graphs”, International Journal of Management and Systems, Vol.21, Jan-Apr, 2005, pp.91-106

[6] Joseph Varghese and A.Antonysamy, Mapana, Journal of Sciences,Vol.8, No.2, July-Dec 2009, pp 7-19