An Algorithm on the Decomposition of Some Complete Tripartite Graphs

Transcription

1 International Journal of Computational Science and Mathematics. ISSN Volume 4, Number 3 (2012), pp International Research Publication House 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, , Tamil Nadu, India 2 HOD, Dept. of Mathematics, Narayanaguru College of Engineering, Tamil Nadu, India franklinsheela@yahoo.co.in n.gnanadhas@gmail.com Abstract Let G= (V, E) be a finite connected simple graph and {G i /i=1,2,n} be a collection of edge-disjoint sub graphs of G such that E(G)=E(G 1 )UE(G 2 )U.UE(G n ), then the collection {G i } is called a decomposition of G. if each G i is connected and E(G i ) =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 K 1,3,m K 2,3,m, K 2,5,m and K 3,5,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 n N, denoted by K n, is a connected simple graph with every vertex is connected with every other vertex by an edge. A graph with n vertices v 1,v 2,..v n, where n =3, and edges {v 1,v 2 }, {v 2,v 3 },..{v n-1,v n },{v n,v 1 } is known as a cycle, C n.

2 226 R.Franklin Richard and N.Gnana Dhas For graph terminology we refer to Harary 1 Graph Decomposition Let G=(V,E) be a connected simple graph of order p and size q. If A decomposition (G 1,G 2,..G n ) of G is said to be a continuous monotonic decomposition (CMD) if each G i is connected and E(Gi) =i i N. Alavi 2, introduced Ascending Sub graph Decomposition(ASD) as a decomposition of G into subgraphs G i (not necessarily connected) and is isomorphic to a proper subgraph G i+1.gnanadhas and Paulraj Joseph introduced a new concept known as continuous monotonic decomposition of Graphs 3. A decomposition, {G 1,G 2,G n } 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 K 2,5,m and K 3,5,m Continuous Monotonic Decomposition of a wide variety of graphs had been studied by Gnanadhas and Paulraj Joseph and Navaneetha Krishnan and Nagarajan 5 If a graph G admits a CMD {G 1,G 2,G n } n N if and only if q= (n+1) C 2 The following two results 6, Joseph Varghese and A. Antonysamy, are about particular classes of complete tripartite graphs which accept CMD.

3 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 n N. 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 n N. 6

4 228 R.Franklin Richard and N.Gnana Dhas Theorem 4: A complete tripartite graph K 2,5,m accepts CMD of G 1,G 2,.G 7n+2 and G 1,G 2,G 7n+4 if and only if m=(7n 2 +5n-2)/2 and m=(7n 2 +9n)/2 respectively n N. 6 Theorem 5: A complete tripartite graph K 3,5,m accepts CMD of G 1,G 2,..G (16n-6) if and only if m=16n 2-11n and CMD of G 1, G 2,.. G 16n+5 if and only if m=16n 2 +11n n N. 6 Proof of Theorem 4: Assume that a complete tripartite graph K 2,5,m accepts CMD of G 1,G 2,..G 7n+2 and G 1,G 2,..G 7n+4, n N. We have q(k 2,5,m ) = [m(2+5)+5(m+2)+2(m+5)]/2 m n N. Case 1: when 7n+2 decompositions K 2,5,m accepts CMD of G 1,G 2,..G 7n+2 iff q(k 2,5,m ) = (7n+2)(7n+3)/2, i.e., 7m+10=(7n+2)(7n+3)/2. i.e., 2m=(7n 2 +5n-2) i.e., m= (7n 2 +5n-2)/2. The values of m are 5, 18, 38, 65, 99 Case 2: when 7n+4 decompositions. K 2,5,m accepts CMD of G 1,G 2,..G 7n+4 iff q(k 2,5,m ) = (7n+4)(7n+5)/2, i.e=(7n+4)(7n+5)/2 =10+7m i.e., m= (7n 2 +9n)/2 for n N The values of m are 8,23,45,74,110, Hence a complete tripartite graph K 2,5,m accepts CMD of G 1,G 2,. G 7n+2 if and only if m=(7n 2 +5n-2)/2 and CMD of G 1,G 2,..G 7n+4 if and only if m=(4n 2 +5n)/2 n N. Conversely Suppose that K 2,5,m is a complete tripartite graph with m=(7n 2 +5n-2)/2 and m=(7n 2 +9n)/2 n N. We know that q(k 2,5,m ) = 10+7m Case 1: when m=(7n 2 +5n-2)/2 q(k 2,5,m ) = 10+7m = 10+7(7n 2 +5n-2)/2 =(6+49n 2 +35n)/2 =(7n+2)(7n+3)/2 Which is of the form k(k+1)/2 n N This implies that K 2,5,m is a connected simple graph, can be decomposed into G 1,G 2,G k n N. i.e., K 2,5,m can be decomposed onto G 1,G 2,.G 7n+2 for n N.

5 An Algorithm on the Decomposition of Some Complete Tripartite Graphs 229 Case 2: when m=(7n 2 +9n)/2 q(k 2,5,m ) = 10+7m = 10+7(7n 2 +9n)/2 =(20+49n 2 +63n)/2 =(7n+4)(7n+5)/2 Which is of the form k(k+1)/2 n N This implies that K 2,5,m is a connected simple graph, can be decomposed into G 1,G 2,.G k n N. i.e., K 2,5,m can be decomposed onto H 1,H 2,.H 7n+4 for n N 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= (4n 2 +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= (5n 2 +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

6 230 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= (7n 2 +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= (16n 2 +11n) (vii)calculate x=16n+5 (vii)calculate y=8m+15 (ix)print m, y and x Step 4 (i)calculate m= (16n 2-11n) (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 H 1 H H 1 H H 1 H H 1 H 18

7 An Algorithm on the Decomposition of Some Complete Tripartite Graphs H 1 H H 1 H H 1 H H 1 H H 1 H H 1 H 42 (2): Output: (K 2,3,m) Enter the value of n 10 a=2,b=3 Sl.No M Q CMD H 1 H H 1 H H 1 H H 1 H H 1 H H 1 H H 1 H H 1 H H 1 H H 1 H 28 (3): Output: K 2,5,m Enter the value of n 10 a=2,b=5 Sl.No M q CMD H 1 H H 1 H H 1 H H 1 H H 1 H H 1 H H 1 H H 1 H H 1 H H 1 H 39

8 232 R.Franklin Richard and N.Gnana Dhas (4): Output: ( K 3,5,m ) Enter the value of n 10 a=3, b=5 Sl.No M q CMD H 1 H H 1 H H 1 H H 1 H H 1 H H 1 H H 1 H H 1 H H 1 H H 1 H 85 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 [4] N.Gnanadhas and J. Paulraj Joseph, Continuous Monotonic Decomposition of Cycles, International Journal of Management and Systems, Vol.19, Jan-Apr, 2003, pp [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 [6] Joseph Varghese and A.Antonysamy, Mapana, Journal of Sciences,Vol.8, No.2, July-Dec 2009, pp 7-19