Vertex Odd Divisor Cordial Labeling for Vertex Switching of Special Graphs

Transcription

1 Global Journal of Pure and Applied Mathematics. ISSN Volume 13, Number 9 (017), pp Research India Publications Vertex Odd Divisor Cordial Labeling for Vertex Switching of Special Graphs G. V. Ghodasara H. & H. B. Kotak Institute of Science, Rajkot, Gujarat - INDIA. D. G. Adalja Marwadi Education Foundation, Rajkot, Gujarat - INDIA Abstract A vertex odd divisor cordial labeling of a graph G with vertex set V is a bijection f : V (G) 1, 3,...,n 1} such that an edge e = uv is assigned label 1 if f (u) f(v) or f(v) f (u) and label 0 otherwise, then e f (0) e f (1) 1. A graph which admits vertex odd divisor cordial labeling is called vertex odd divisor cordial graph. In this paper we prove that vertex switching of cycle C n (n 4) with one chord, cycle C n (n 5) with twin chords, cycle C n (n 6) with triangle, gear graph, shell graph, flower graph are vertex odd divisor cordial. AMS subject classification: 05C78. Keywords: Vertex odd divisor cordial graph, Vertex switching of a graph. 1. Introduction The graphs considered here are finite, simple and undirected graph. For terms not defined here, we refer to Gross and Yellen [3]. For standard terminology and notations related to number theory we refer to Burton [1]. The most recent findings on various graph labeling techniques can be found in Gallian []. Varatharajan et al. introduced the concept of divisor cordial labeling of graphs.

2 556 G. V. Ghodasara and D. G. Adalja Definition 1.1. [8] Let G = (V, E) be a graph, f : V (G) 1,,..., V (G) } be a bijection and induced function f : E(G) 0, 1} be defined as f 1; if f (u) f(v) or f(v) f (u). (e = uv) = 0; otherwise. Then f is called divisor cordial labeling if e f (0) e f (1) 1. A graph which admits divisor cordial labeling is called divisor cordial graph. Definition 1.. [6] The vertex switching G v of a graph G is the graph obtained by taking a vertex v of G, removing all the edges incident to v and adding edges joining v to every other vertex which is not adjacent to v in G. Throughout this paper V (G) and E(G) denote the cardinality of vertex set and edge set of graph G respectively.. Vertex odd divisor cordial labeling Muthaiyan and Pugalenthi introduced the concept of vertex odd divisor cordial labeling of graphs. Definition.1. [4] Let G = (V, E) be a simple graph, f : V (G) 1, 3,...,n 1} be a bijection and induced function f : E(G) 0, 1} be defined as f 1; if f (u) f(v) or f(v) f (u). (e = uv) = 0; otherwise. Then f is called vertex odd divisor cordial labeling if e f (0) e f (1) 1. A graph which admits vertex odd divisor cordial labeling is called vertex odd divisor cordial graph. The divisor cordial and vertex odd divisor cordial labeling of various types of graphs are presented in [6, 9, 5]. 3. Vertex odd divisor cordial labeling for vertex switching of cycle related graphs Definition 3.1. [] A chord of a cycle C n is an edge joining two non-adjacent vertices. Theorem 3.. The graph obtained by vertex switching of cycle C n (n 4) with one chord is vertex odd divisor cordial, where chord forms a triangle with two edges of C n. Proof. Let G be the cycle C n with one chord. Let v 1,v,...,v n be the successive vertices of C n and e = v v n be the chord of C n. The edges e = v v n,e 1 = v 1 v,e = v 1 v n form

3 Vertex Odd Divisor Cordial Labeling for Vertex Switching of Special Graphs 557 a triangle. Without loss of generality let the switched vertex be v 1 (of degree or degree 3) and let G v1 denote the vertex switching of G with respect to v 1. V(G v1 ) =n. We define labeling function f : V(G v1 ) 1, 3, 5,...,n 1} as per the following cases. Case 1: deg(v 1 ) =. In this case E(G v1 ) =n 4. Here we define labeling f as f(v i ) = i 1; 1 i 4. i + 1; 5 i n 1. f(v n ) = 9. If label of v n 1 is a multiple of 9 then interchange the labels of v n 1 and v n. In view of the above labeling pattern, we have e f (1) = e f (0) = n. Case : deg(v 1 ) = 3. In this case E(G v1 ) =n 6. Here we define labeling f as f(v i ) = i 1; i = 1,, 6 i n. i 3; 4 i 5. f(v 3 ) = 9. In view of the above labeling pattern we have e f (1) = e f (0) = n 3. Thus e f (0) e f (1) 1. Hence vertex switching of cycle C n with one chord is a vertex odd divisor cordial graph. Example 3.3. Cycle graph C 7 with one chord and Vertex odd divisor cordial labeling for the graph obtained by vertex switching of cycle C 7 with one chord with respect to vertex of degree and degree 3 respectively is shown in Figure 1. Figure 1: Definition 3.4. [] Two chords of a cycle C n are said to be twin chords if they form a triangle with an edge of C n. For positive integers n and p with 5 p + n, C n,p

4 558 G. V. Ghodasara and D. G. Adalja is the graph consisting of a cycle C n with twin chords where chords form cycles C p,c 3 and C n+1 p without chords the edges of C n. Theorem 3.5. The graph obtained by vertex switching of cycle with twin chords C n,3 (n 5) is vertex odd divisor cordial. Proof. Let V(C n,3 ) =v 1,v,...,v n }, e 1 = v n v, e = v n v 3 be the chords. Without loss of generality let v 1 be the switched vertex and let (C n,3 ) v1 denote the vertex switching of C n,3 with respect to vertex v 1. V ((C n,3 ) v1 ) =n. We define labeling function f : V ((C n,3 ) v1 ) 1, 3, 5,...,n 1} as per the following cases. Case 1: deg(v 1 ) =. In this case E((C n,3 ) v1 ) =n 3. Here we define labeling f as f(v i ) = i 1; 1 i 4. i + 1; 5 i n 1. f(v n ) = 9. If label of v n 1 is a multiple of 9 then interchange the labels of v n 1 and v n. In view of the above labeling pattern we have e f (1) = n 1,e f (0) = n. Case : deg(v 1 ) = 3. In this case E((C n,3 ) v1 ) =n 5. Here we define labeling f as f(v i ) = i 1; i = 1,, 6 i n. i 3; 4 i 5. f(v 3 ) = 9. If label of v n 1 is a multiple of 3 then interchange the labels of v n 1 and v n. In view n 5 n 5 of the above labeling pattern we have e f (1) =,e f (0) =. Case 3: deg(v 1 ) = 4. In this case E((C n,3 ) v1 ) =n 7. Here we define labeling f as f(v i ) = i 1; i = 1,, 6 i n. i 3; 4 i 5. f(v 3 ) = 9. In view of the above labeling pattern we have the following. n 7 e f (1) =. n 7 e f (0) =.

5 Vertex Odd Divisor Cordial Labeling for Vertex Switching of Special Graphs 559 Hence vertex switching of C n,3 is a vertex odd divisor cordial graph. Example 3.6. C 8,3 graph and Vertex odd divisor cordial labeling for the graph obtained by vertex switching of C 8,3 with respect to vertex of degree, degree 3 and degree 4 respectively is shown in Figure. Figure : Definition 3.7. [] 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+, C q+, C r+ without chords. Theorem 3.8. The graph obtained by vertex switching of cycle with triangle C n (1, 1,n 5)(n 8) is vertex odd divisor cordial. Proof. Let v 1,v,...,v n be the successive vertices of cycle with triangle G = C n (1, 1,n 5) and e 1 = v 1 v n 1,e = v 1 v 3,e 3 = v n 1 v 3 be the chords of G. Without loss of generality let v 1 be the switched vertex and let G v1 denote the vertex switching of G with respect to v 1. V(G v1 ) =n. We define labeling function f : V(G v1 ) 1, 3, 5,...,n 1} as per the following cases. Case 1: deg(v 1 ) =. In this case E(G v1 ) =n. Here we define labeling f as i 1; i = 1,, 6 i 7. f(v i ) = i 3; 4 i 5. i + 1; 8 i n 3. f(v 3 ) = 9. f(v n ) = 15. f(v n 1 ) = p, f(v n ) = p 1, where p 1 and p are the largest prime number and previous largest prime number respectively. In view of the above labeling pattern we have e f (1) = e f (0) = n 1.

6 5530 G. V. Ghodasara and D. G. Adalja Case : deg(v 1 ) = 4. In this case E(G v1 ) =n 6. f(v i ) = i 1; i = 1, 6 i 7. i 3; 3 i 5. i + 1; 8 i n. f(v ) = 9. f(v n 1 ) = 15. f(v n ) = p, where p is the largest prime number. In view of the above labeling pattern we have e f (1) = e f (0) = n 3. Thus e f (0) e f (1) 1. Hence vertex switching of C n (1, 1,n 5) is a vertex odd divisor cordial graph. Example 3.9. C 8 (1, 1, 3) graph and Vertex odd divisor cordial labeling for the graph obtained by vertex switching of C 8 (1, 1, 3) with respect to vertex of degree and degree 4 respectively is shown in Figure 3. Figure 3: 4. Vertex odd divisor cordial labeling for vertex switching of wheel and shell related graphs Definition 4.1. [3] The wheel graph W n is the join of C n and K 1 i.e. W n = C n +K 1. Here the edges(vertices) of C n are called rim edges(rim vertices) and the vertex corresponding to K 1 is called apex. Theorem 4.. The graph obtained by vertex switching of any rim vertex of wheel W n is vertex odd divisor cordial. Proof. Let v 0 be the apex and v 1,v,...,v n be the rim vertices of wheel W n. Let (W n ) v1 denote the graph obtained by switching of a rim vertex v 1 of W n. Here V ((W n ) v1 ) = n + 1 and E((W n ) v1 ) =3n 6. We define labeling function f : V ((W n ) v1 ) 1, 3, 5,...,n + 1} as

7 Vertex Odd Divisor Cordial Labeling for Vertex Switching of Special Graphs 5531 For n 9 : For n>9 : f(v 0 ) = 1. f(v i ) = i + 1; 1 i n. f(v 0 ) = 1. f(v 1 ) = 3. f(v i ) = p i+1 ; 1 i k. ( n 6 f(v i+1 ) = 3f(v i ); 1 i k, where k = 6 ). and p i+1 = i + 1 th prime number. For the remaining vertices v k+,v k+3,...,v n 1, assign the vertex labels such that for any pair of adjacent vertices (v i,v j ), f(v i ) f(v j ) and f(v j ) f(v i ). In view of the above labeling pattern we have when n is even: e f (1) = 3n 6 = e f (0). 3n 6 e f (1) =. when n is odd: 3n 6 e f (0) =. Thus e f (0) e f (1) 1. Hence vertex switching of any rim vertex of W n is a vertex odd divisor cordial graph. Example 4.3. Wheel graph W 9 and Vertex odd divisor cordial labeling of the graph obtained by vertex switching of a rim vertex v 1 of wheel W 9 is shown in Figure 4. Figure 4: Definition 4.4. [3] The gear graph G n is obtained from the wheel W n by subdividing every rim edge of W n. Theorem 4.5. The graph obtained by vertex switching of gear graph G n (except apex vertex) is vertex odd divisor cordial.

8 553 G. V. Ghodasara and D. G. Adalja Proof. Let v 0 be the apex vertex and v 1,v,...,v n be other vertices of gear graph G n, where ; if i is even. deg(v i ) = 3; if i is odd. Now the graph obtained by vertex switching of rim vertices v i and v j of degree are isomorphic to each other, i, j. Similarly the graph obtained by vertex switching of rim vertices v i and v j of degree 3 are isomorphic to each other i, j. Hence we require to discuss two cases. (i) Vertex switching of an arbitrary vertex say v 1 of G n of degree 3. (ii) Vertex switching of an arbitrary vertex say v of G n of degree. Let (G n )v i denote the vertex switching of G n with respect to the vertex v i,i = 1,. Case 1: Vertex switching of G n with respect to vertex v 1 (of degree 3). Here V(G n ) = n + 1 and E(G n ) =5n 6. We define labeling function f : V ((G n )v 1 ) 1, 3,...,4n + 1} as follows. Our aim is to generate 5n 6 edges with label 1 and 5n 6 f(v 1 ) = 1, which generates n 3 edges with label 1 and f(v 0 ) = 3. edges with label 0. Let Now it remains to generate k = 5n 6 (n 3) edges with label 1. For the vertices v 3,v 5,...,v k assign the vertex label as per following ordered pattern upto it generate k edges with label 1. f(v i+1 ) = 3(i + 1); 1 i k, n ; if n is even. where k = n + 1 ; if n is odd. For the remaining vertices v k+1,v k+,...,v n 1,v,v 4,...,v n of G n, assign the vertex labels such that for any pair of adjacent vertices (v i,v j ), f(v i ) f(v j ) and f(v j ) f(v i ). In view of above defined labeling pattern we have, when n is even: e f (1) = 5n 6 = e f (0).

9 Vertex Odd Divisor Cordial Labeling for Vertex Switching of Special Graphs 5533 Thus e f (0) e f (1) 1. when n is odd: e f (1) = 5n 5. e f (0) = 5n 7. Case : Vertex switching of G n with respect to vertex v (of degree ). Here V(G n ) = n + 1 and E(G n ) =5n 4. We follow the same labeling pattern as in Case 1. Then we have, when n is even: e f (1) = 5n 4 = e f (0). e f (1) = 5n 3. when n is odd: e f (0) = 5n 5. Hence e f (0) e f (1) 1. Thus the graph obtained by vertex switching of any vertex of G n is vertex odd divisor cordial. Example 4.6. Gear graph G 6 and Vertex odd divisor cordial labeling for the graph obtained by vertex switching of vertices of degree 3 and degree respectively of gear graph G 6 is shown in Figure 5. Figure 5: Definition 4.7. [3] The shell graph S n is obtained by taking n 3 concurrent chords in cycle C n. The vertex at which all the chords are concurrent is called the apex vertex. Theorem 4.8. The graph obtained by vertex switching of any vertex (except apex vertex) of shell graph S n is vertex odd divisor cordial. Proof. Let u 0 be the apex vertex and u 1,u,...,u n 1 be the other vertices of shell S n, where deg(u 1 ) ==deg(u n 1 ) and deg(u i ) = 3,i =, 3,...,n. The graphs obtained by vertex switching of vertices of same degree are isomorphic to each other. Hence we require to discuss following two cases: (i) Vertex switching of an arbitrary vertex say u of S n of degree 3.

10 5534 G. V. Ghodasara and D. G. Adalja (ii) Vertex switching of an arbitrary vertex say u 1 of S n of degree. Let (S n ) ui denote the vertex switching of S n with respect to the vertex u i of S n, i = 1,. Case 1: Vertex switching of u, where deg(u ) = 3. Here V ((S n ) u ) = n and E((S n ) u ) =3n 10. We define labeling function f : V ((S n ) u ) 1, 3,...n 1} as follows. For n 15 : For n>15 : f(u 0 ) = 1. f(u 1 ) = p, where p is the largest prime number. f(u ) = 3. f(u i ) = i 1; 3 i n 1. f(u 0 ) = 1. f(u 1 ) = p, where p is the largest prime number. f(u ) = 3. f(u i 1 ) = p i ; i k. ( n 10 f(u i ) = 3f(u i 1 ); i k, where k = 6 ). and p i = i th prime number. For the remaining vertices u k+1,u k+,u k+3,u k+4,..., u n 1, assign the vertex labels such that for any pair of adjacent vertices (u i,u j ), f(u i ) f(u j ) and f(u j ) f(u i ). In view of above labeling pattern we have, Thus e f (0) e f (1) 1. 3n 10 when n is even: e f (1) = = e f (0). e f (1) = 3n 9. when n is odd: 3n 11 e f (0) =. Case : Vertex switching of u 1, where deg(u 1 ) =. Here V ((S n ) u1 ) =n, E((S n ) u1 ) = 3n 8. We define labeling function f : V ((S n ) u1 ) 1, 3,...n 1} as follows. For n 9 : f(u i ) = i + 1; 0 i n 1.

11 Vertex Odd Divisor Cordial Labeling for Vertex Switching of Special Graphs 5535 For n>9 : f(u 0 ) = 1, f(u 1 ) = 3, f(u i ) = p i+1 ; 1 i k, n 4 f(u i+1 ) = 3f(u i ); 1 i k, where k = 6 and p i+1 = i + 1 th prime number. For the remaining vertices u k+,u k+3,...,u n 1 of (S n ) u1, assign the vertex labels such that for any pair of adjacent vertices (u i,u j ), f(u i ) f(u j ) and f(u j ) f(u i ). In view of above labeling pattern we have, when n is even: e f (1) = 3n 8 = e f (0). e f (1) = 3n 7. when n is odd: e f (0) = 3n 9. Thus e f (0) e f (1) 1. Thus the graph obtained by vertex switching of any vertex (except apex vertex) of S n is vertex odd divisor cordial. Example 4.9. Shell graph S 7 and Vertex odd divisor cordial labeling for the graph obtained by switching of vertex of degree 3 and degree respetively of S 7 is shown in Figure 6.. Figure 6: Definition [3] The flower graph fl n (n 3) is obtained from a helm H n by joining each pendant vertex to the apex vertex of H n. Here we consider rim vertices as internal vertices.

12 5536 G. V. Ghodasara and D. G. Adalja Theorem The graph obtained by vertex switching of any vertex of flower graph fl n is vertex odd divisor cordial. Proof. Let u 0 be the apex vertex of helm H n. Let u 1,u,...,u n be the rim vertices of wheel W n in H n. Let e 1,e,...,e n be the spoke edges where e i = u i v i,i = 1,,...,n, deg(u i ) = 4 and deg(v i ) =. Then flower graph fl n is obtained by joining the vertices v 1,v,...,v n to apex vertex u 0 of H n. Now the graph obtained by vertex switching of internal vertices u i and u j are isomorphic to each other, i, j. Similarly the graph obtained by vertex switching of external vertices u i and u j are isomorphic to each other i, j. Hence it is required to discuss following two cases. (i) Vertex switching of an internal vertex u 1 of degree 4. (ii) Vertex switching of external vertex say v 1 of degree. Let (f l n ) u1 and (f l n ) v1 denote the vertex switching of fl n with respect to the vertex u 1 and v 1 respectively. Case 1: Vertex switching of an internal vertex say u 1 of fl n of degree 4. Here V ((f l n ) u1 ) = n + 1, E((f l n ) u1 ) = 6n 8. We define labeling function f : V ((f l n ) u1 ) 1, 3,...4n + 1} as follows. For n 8 : f(u 0 ) = 1. f(u 1 ) = 3. f(v 1 ) = p, where p is the largest prime number. Assign the remaining labels to the remaining vertices in any order such that for any pair of adjacent vertices (u i,u j ) and (u i,v j ), f(u i ) f(u j ) and f(u i ) f(v j ). For n>8 : f(u 0 ) = 1. f(u 1 ) = 3. f(v 1 ) = p, where p is the largest prime number. n f(u i ) = p i ; i k, where k =. 3 n f(v i ) = 3f(u i ); i k, where k =. 3 and p i = i th prime number. Assign the remaining labels to remaining vertices u k+1,u k+,...,u n,v k+1,v k+,...,v n of (f l n ) u1 such that f(u i ) f(u i+1 ); k + 1 i n 1. f(u i ) f(v i ); k + 1 i n.

13 Vertex Odd Divisor Cordial Labeling for Vertex Switching of Special Graphs 5537 In view of above labeling pattern we have, e f (1) = 3n 4 = e f (0). Case : Vertex switching of an external vertex say v 1 of fl n of degree. Here V ((f l n ) v1 ) = n + 1, E((f l n ) v1 ) = 6n 4. We define labeling function f : V ((f l n ) v1 ) 1, 3,...4n + 1} as follows. f(u 0 ) = 1, f(v 1 ) = 3, n f(u i ) = p i ; i k, where k =. 3 f(v i ) = 3f(u i ); i k, where k = n 3. and p i = i th prime number. Assign the remaining labels to remaining vertices u k+1,u k+,...,u n,v k+1,v k+,...,v n of (f l n ) v1 such that In view of above labeling pattern we have f(u i ) f(u i+1 ); k + 1 i n 1. f(u i ) f(v i ); k + 1 i n. e f (1) = 3n = e f (0). Hence the graph obtained by vertex switching of any vertex of fl n (except apex vertex) is vertex odd divisor cordial graph. Example 4.1. Flower graph fl 4 and Vertex odd divisor cordial labeling of its vertex switching with respect to vertex of degree and degree 4 are shown in Figure 7. Figure 7: 5. Concluing Remarks It is already proved that cycle with one chord, cycle with twin chord, cycle with triangle, wheel graph, gear graph, shell and flower graph are vertex odd divisor cordial. We have

14 5538 G. V. Ghodasara and D. G. Adalja discussed vertex odd divisor cordiality of these graphs in context of vertex switching of any arbitrary vertex (except apex vertex). Hence it is concluded that vertex odd divisor cordial labeling is preserved under the opertion vertex switching of the above graphs. To examine the same for other graphs is an open area of research. References [1] D. M. Burton, Elementary Number Theory, Brown Publishers, Second Edition, (1990). [] J. A. Gallian, A Dynamic Survey of Graph Labeling, The Electronic Journal of Combinatorics, (016), # DS6. [3] J. Gross and J. Yellen, Graph Theory and Its Applications, CRC Press, (1999). Indian Acad. Math., 7 () (005) [4] A. Muthaiyan and P. Pugalenthi, Vertex Odd Divisor Cordial Graphs, Asia Pacific Journal of Research, Vol. 1, (015). [5] A. Muthaiyan and P. Pugalenthi, Vertex Odd Divisor Cordial Graphs, International Journal of Innovative Science, Eng. and Tech., (10) (015). [6] S. K. Vaidya and N. H. Shah, Further Results on Divisor Cordial Labeling, Annals of Pure and Applied Mathematics, 4() (013), [7] S. K. Vaidya and K. K. Kanani, Prime Labeling for Some Cycle Related Graphs, ž Journal of Mathematics Rsearch, Vol., No., 010, pp [8] R. Varatharajan, S. Navanaeethakrishnan and K. Nagarajan, Divisor Cordial Graphs, International J. Math. Combin., 4 (011) [9] R. Varatharajan, S. Navanaeethakrishnan and K. Nagarajan, Special Classes of Divisor Cordial Graphs, International Mathematical Forum, 7(35) (01), [10] G. V. Ghodasara, D. G. Adalja, Divisor Cordial Labeling in Context of Ring Sum of Graphs, International Journal of Mathematics and Soft Computing, Vol. 7, No. 1, 017, pp [11] G. V. Ghodasara, D. G. Adalja, Divisor Cordial Labeling for Vertex Switching and Duplication of Special Graphs, International Journal of Mathematics And its Applications, Vol. 4, Issue 3B, 016, pp