PAijpam.eu PRIME CORDIAL LABELING OF THE GRAPHS RELATED TO CYCLE WITH ONE CHORD, TWIN CHORDS AND TRIANGLE G.V. Ghodasara 1, J.P.

Transcription

1 International Journal of Pure and Applied Mathematics Volume 89 No , ISSN: (printed version); ISSN: (on-line version) url: doi: PAijpam.eu PRIME CORDIAL LABELING OF THE GRAPHS RELATED TO CYCLE WITH ONE CHORD, TWIN CHORDS AND TRIANGLE G.V. Ghodasara 1, J.P. Jena 2 1 H.&H.B. Kotak Institute of Science Rajkot, Gujarat, INDIA 2 L.E. College, Morbi Gujarat, INDIA Abstract: We discuss here prime cordial labeling of cycle C n with one chord, cycle C n with twin chords, cycle C n with triangle. We also discussprimecordial labeling of the graph obtained by joining two copies of cycle with one chord by path of arbitrary length and the graph obtained by joining two copies of cycle with twin chords by path of arbitrary length. AMS Subject Classification: 05C78 Key Words: cycle with one chord, cycle with twin chords, cycle with triangle, prime cordial graph 1. Introduction We begin with simple, finite undirected graph G = (V,E). In the present work C n and P k denotes the cycle with n vertices and path with k vertices respectively. For all other terminology and notations we follow Harary[1]. Definition 1.1. A chord of a cycle C n is an edge joining two non-adjacent vertices of C n. Definition 1.2. Two chords of a cycle C n (n 5) are said to betwin chords if they form a triangle with an edge of the cycle C n. Received: June 24, 2013 Correspondence author c 2013 Academic Publications, Ltd. url:

2 80 G.V. Ghodasara, J.P. Jena For positive integers n and p with 3 p n 2, C n,p is the graph consisting of a cycle C n with a pair of twin chords with which the edges of C n form cycles C p, C 3 and C n+1 p without chords. Definition 1.3. A cycle with triangle is a cycle with three chords which by themselves form a triangle. For positive integers p,q,r and n 6 with p + q + r + 3 = n, C n (p,q,r) denotes a cycle with triangle whose edges form the edges of cycles C p+2, C q+2, C r+2 without chords. Definition 1.4. If the vertices of the graph are assigned values subject to certain conditions is known as graph labeling. Detailed survey on graph labeling is given and updated by Gallian[2]. Definition 1.5. A prime cordial labeling of a graph G is a bijection f : V(G) {1,2,..., V(G) } and the induced edge function f : V(G) {0,1} is defined by f (e = uv) = 1; if gcd(f(u),f(v)) = 1 = 0; if gcd(f(u),f(v)) > 1 satisfies the condition e f (0) e f (1) 1 where e f (0), e f (1) be the number of edges with labels 0 and 1 respectively under f. Sundaram et al.[3] have introduced the notion of prime cordial labeling. In same paper they discussed prime cordial labeling of cycles C n, paths P n, K 1,n etc. Vaidya and Vihol[5] proved that the graph obtained by joining two copies of a fixed cycle by a path of arbitrary length is prime cordial. In [5] and [6] same authors have discussed prime cordial labeling of various graphs. Vaidya and Shah[7] have discussed prime cordial labeling of split graphs and middle graphs of different graphs. In this paper we prove that cycle C n with one chord, C n with twin chords and C n with triangle are prime cordial for all n. We also prove that the graph obtained by joining two copies of cycle with one chord by path of arbitrary length and the graph obtained by joining two copies of cycle with twin chords by path of arbitrary length. 2. Main Results Theorem 2.1 Cycle C n with one chord is prime cordial for all n.

3 PRIME CORDIAL LABELING OF THE GRAPHS RELATED Proof. Let G be the cycle C n with one chord. Let u 1,u 2,...,u n be the vertices of cycle C n and let e = u 1 u 3 be the chord of cycle C n. We define the labeling f : V(G) {1,2,...,n} as follows. Case 1. n is even. Then f(u 1 ) = 2, f(u 2 ) = 4, f(u 3 ) = 6, f(u 4 ) = 3. Label the consecutive vertices of u i, where 5 i n 2 + 3, by labels 1,5,7,9,...,n 1respectively andlabeltheremainingvertices u i, where n 2 +4 i n, by labels 8,10,...,n respectively. One can observe that the labeling defined above satisfies the conditions of prime cordial labeling and the graph under consideration is prime cordial graph in this case. Case 2. n is odd. Then f(u 1 ) = 2, f(u 2 ) = 4, f(u 3 ) = 6, f(u 4 ) = 3. Label the consecutive vertices of u i, where 5 i n , by labels 1,5,7,9,...,n respectively and label the remaining vertices u i, where n i n, by labels 8,10,...,n 1 respectively. One can observe that the labeling defined above satisfies the conditions of prime cordial labeling and the graph under consideration is prime cordial graph in this case. Illustration 2.1. For better understanding of above defined labeling pattern the prime cordial labeling of C 6 with one chord is shown in Figure 1. It is the case related to n is even. Figure 1: Prime cordial labeling of C 6 with one chord Theorem 2.2. Cycle C n with twin chords is prime cordial for all n. Proof. Let G be the cycle C n with twin chords. Let u 1,u 2,...,u n be the vertices of cycle C n. Let e 1 = u 1 u 3 and e 2 = u 1 u 4 be the chords of cycle C n. We define the labeling f : V(G) {1,2,...,n} as follows.

4 82 G.V. Ghodasara, J.P. Jena Case 1. n is even. Then f(u 1 ) = 2, f(u 2 ) = 4, f(u 3 ) = 6, f(u 4 ) = 3. Label the consecutive vertices of u i, where 5 i n 2 + 3, by labels 1,5,7,9,...,n 1respectively andlabeltheremainingvertices u i, where n 2 +4 i n, by labels 8,10,...,n respectively. Case 2. n is odd. f(u 1 ) = 2,f(u 2 ) = 4, f(u 3 ) = 6, f(u 4 ) = 3. Label the consecutive vertices of u i, where 5 i n , by labels 1,5,7,9,...,n respectively and label the remaining vertices u i, where n i n, by labels 8,10,...,n 1 respectively. One can observe that in each case the labeling defined above satisfies the conditions of prime cordial labeling and the graph under consideration is prime cordial graph. Illustration 2.2. For better understanding of above defined labeling pattern the prime cordial labeling of C 6 with twin chords is shown in Figure 2. It is the case related to n is odd. Figure 2: Prime cordial labeling of C 7 with twin chord Theorem 2.3. CycleC n withtriangle C n (1,1,n 2) isprimecordial except n = 7. Proof. For the graph C 7 the possible assignment of labels to adjacent vertices will be (1,2), (1,3), (1,4), (1,5), (1,6), (1,7), (2,3), (2,4), (2,5), (2,6), (2,7), (3,4), (3,5), (3,6), (3,7), (4,5), (4,6), (4,7), (5,6), (5,7), (6,7). Such assignment will generate maximum four edges with label 0 and minimum six edges with label 1.That is, e f (0) e f (1) = 3 > 1. Therefore C 7 is not prime cordial graph. Hence we consider the case when n 7. Let G be the cycle C n with triangle C n (1,1,n 2), n 7. Let u 1,u 2,...,u n be the vertices of cycle C n. Let e 1 = u 1 u 3, e 2 = u 3 u n 1 and e 3 = u 1 u n 1 be the

5 PRIME CORDIAL LABELING OF THE GRAPHS RELATED chords of cycle C n. We define the labeling f : V(G) {1,2,...,n} as follows. Case 1. n is even. Then f(u 1 ) = 2, f(u 2 ) = 4, f(u 3 ) = 6, f(u n ) = 1. Label the consecutive vertices of u i, where 4 i n 2 + 2, by labels 3,5,7,9,...,n 1respectively andlabeltheremainingvertices u i, where n 2 +3 i n 1, by labels 8,10,...,n respectively. Case 2. n is odd, n 7. Then f(u 1 ) = 2,f(u 2 ) = 4, f(u 3 ) = 6, f(u n ) = 1. Label the consecutive vertices of u i, where 4 i n by labels 3,5,7,9,...,n respectively and label the remaining vertices u i, where n i n 1 by labels 8,10,...,n 1 respectively. One can observe that in each case the labeling defined above satisfies the conditions of prime cordial labeling and the graph under consideration is prime cordial graph. Illustration 2.3. For better understanding of above defined labeling pattern the prime cordial labeling of C 6 with triangle is shown in Figure 3. It is the case related to n is even. Figure 3: Prime cordial labeling of C 6 with triangle Theorem 2.4. The graph G obtained by joining two copies of cycle with one chord by a path of arbitrary length is prime cordial. Proof. Let G be the graph obtained by joining two copies of cycle C n with one chord by path P k. Let u 1,u 2,...,u n be the vertices of first copy of cycle with one chord, w 1,w 2...,w n be the vertices of second copy of cycle with one chord and v 1,v 2,...,v k be the vertices of path P k with v 1 = u 1 and v k = w 1. Let e = u 1 u 3 be the chord in first copy of cycle C n and e = w 1 w 3 be the

6 84 G.V. Ghodasara, J.P. Jena chord in second copy of cycle C n. To define labeling function f : V(G) {1,2,...,2n+k 2} we consider following cases. Case 1. n = 4, k is odd. Let k = 2t+1,t N. f(u 1 ) = 3,f(u 2 ) = 1, f(u 3 ) = 5, f(u 4 ) = 7 f(v 1 ) = f(u 1 ) = 3; f(v i ) = 2n+2i 3; if 2 i t+1. = 2{i (t+1)}; if t+2 i 2t+1. Case 2. n = 4, k is even. Let k = 2t,t N. f(u 1 ) = 3,f(u 2 ) = 1, f(u 3 ) = 5, f(u 4 ) = 7 f(v 1 ) = f(u 1 ) = 3; f(v i ) = 2n+2i 3; if 2 i t. = 2(i t); if t+1 i 2t. Case 3. n 5, k is odd. Let k = 2t+1,t N. f(u 1 ) = 1, f(u 2 ) = 3, f(u 3 ) = 9 f(u i ) = 2i 3; if 4 i 5 = 2i 1; if 6 i n. f(v 1 ) = f(u 1 ) = 1; f(v i ) = 2n+2i 3; if 2 i t+1. = 2{i (t+1)}; if t+2 i 2t+1. Case 4. n 5, k is even. Let k = 2t,t N. f(u 1 ) = 1, f(u 2 ) = 3, f(u 3 ) = 9 f(u i ) = 2i 3; if 4 i 5 = 2i 1; if 6 i n. f(v 1 ) = f(u 1 ) = 1; f(v i ) = 2n+2i 3; if 2 i t. = 2{i t}; if t+1 i 2t. One can observe that in each case the labeling defined above satisfies the conditions of prime cordial labeling and the graph under consideration is prime cordial graph.

7 PRIME CORDIAL LABELING OF THE GRAPHS RELATED Illustration 2.4. For better understanding of above defined labeling pattern the prime cordial labeling of graph obtained by joining two copies of C 6 with one chord by path P 7 is shown in Figure 4. It is the case related to n 5 and k is odd. Figure 4: Prime cordial labeling of graph obtained by joining two copies of C 6 with one chord by path P 7 Theorem 2.5. The graph G obtained by joining two copies of cycle with twin chords by a path of arbitrary length is prime cordial. Proof. Let G be the graph obtained by joining two copies of cycle C n with twin chords by path P k. Let u 1,u 2,...,u n be the vertices of first copy of cycle with twin chords, w 1,w 2,...,w n be the vertices of second copy of cycle with twin chords and v 1,v 2,...,v k be the vertices of path P k with v 1 = u 1 and v k = w 1. Let e 1 = u 1 u 3 and e 2 = u 1 u 4 be the chords in first copy of cycle C n and e 1 = w 1w 3 and e 2 = w 1w 4 be the chords in second copy of cycle C n. We define labeling function f : V(G) {1,2,...,2n+k 2} as follows. Case 1. k is odd. Let k = 2t+1,t N. f(u 1 ) = 1, f(u 2 ) = 3, f(u 3 ) = 9 f(u i ) = 2i 3; if 4 i 5 = 2i 1; if 6 i n. f(v 1 ) = f(u 1 ) = 1; f(v i ) = 2n+2i 3; if 2 i t+1. = 2{i (t+1)}; if t+2 i 2t+1. Case 2. k is even. Let k = 2t,t N. f(u 1 ) = 1, f(u 2 ) = 3, f(u 3 ) = 9 f(u i ) = 2i 3; if 4 i 5 = 2i 1; if 6 i n.

8 86 G.V. Ghodasara, J.P. Jena f(v 1 ) = f(u 1 ) = 1; f(v i ) = 2n+2i 3; if 2 i t. = 2(i t); if t+1 i 2t. One can observe that in each case the labeling defined above satisfies the conditions of prime cordial labeling and the graph under consideration is prime cordial graph. Illustration 2.5. For well understanding of above defined labeling pattern the prime cordial labeling of graph obtained by joining two copies of C 6 with twin chords by path P 8 is shown in Figure 5. It is the case related to n 5 and k is even. Figure 5: Prime cordial labeling of graph obtained by joining two copies of C 6 with twin chords by path P 8 3. Conclusion In this paper we investigated five new prime cordial graphs. All the results in this paper are novel. For the better understanding of the proofs of the theorems, labeling pattern defined in each theorem is demonstrated by illustration. References [1] F. Harary, Graph Theory, Addision-wesley, Reading, MA (1969). [2] J.A. Gallian, A dynamic survey of graph labeling, The Electronics Journal of Combinatorics, 19 (2012), [3] M. Sundaram, R. Ponraj, S. Somasundram, Prime cordial labeling of graphs, Journal of Indian Acadamy of Mathematics, 27 (2005),

9 PRIME CORDIAL LABELING OF THE GRAPHS RELATED [4] M.A. Seoud, M.A. Salim, Two upper bounds of prime cordial graphs, Journal of Combinatorial Mathematics and Combinatorial Computing, 75 (2010), [5] S.K. Vaidya, P.L. Vihol, Prime cordial labeling for some cycle related graphs, International Journal of Open Problems in Computure Science Mathematics, 3, No. 5 (2010), [6] S.K. Vaidya, P.L. Vihol, Prime cordial labeling for some graphs, Modern Applied Science, 4, No. 8 (2010), [7] S.K. Vaidya, N.H. Shah, Prime cordial labeling of some graphs, Open Journal of Discrete Mathematics, 2, No. 1 (2012), 11-16, doi: /ojdm

10 88