Divisor Cordial Labeling in the Context of Graph Operations on Bistar

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1 Global Journal of Pure and Applied Mathematics. ISSN Volume 12, Number 3 (2016), pp Research India Publications Divisor Cordial Labeling in the Context of Graph Operations on Bistar M. I. Bosmia Government Engineering College, Sector-28, Gandhinagar, Gujarat, INDIA. K. K. Kanani Government Engineering College, Rajkot, Gujarat, INDIA. Abstract A divisor cordial labeling of a graph G with vertex set V (G) is a bijection f from V (G) to {1, 2,..., V (G) } such that an edge e = uv is assigned the label 1 if f (u) f(v) or f(v) f (u) and the label 0 otherwise, then e f (0) e f (1) 1. A graph which admits a divisor cordial labeling is called a divisor cordial graph. In this paper we prove that bistar B m,n, splitting graph of bistar B m,n, degree splitting graph of bistar B m,n, shadow graph of bistar B m,n, restricted square graph of bistar B m,n, barycentric subdivision of bistar B m,n and corona product of bistar B m,n with K 1 admit divisor cordial labeling. AMS subject classification: 05C78. Keywords: Divisor Cordial Labeling, Bistar. 1. Introduction Throughout this work, by a graph we mean finite, connected, undirected, simple graph G = (V (G), E(G)) of order V (G) and size E(G). For any undefined notations and terminology we follow Gross and Yellen [4] while for number theory we follow Burton [1]. Definition 1.1. If the vertices are assigned values subject to certain condition(s) then it is known as graph labeling. A dynamic survey on various graph labeling technique is regularly updated by Gallian [3].

2 2606 M. I. Bosmia and K. K. Kanani Definition 1.2. A mapping f : V (G) {0, 1} is called binary vertex labeling of G and f(v)is called the label of the vertex v of G under f. Notation 1.3. If for an edge e = uv, the induced edge labeling f : E(G) {0, 1} is given by f (e = uv) = f (u) f(v). Then v f (i) = number of vertices of G having label i under f, e f (i) = number of edges of G having label i under f. Definition 1.4. A binary vertex labeling f of a graph G is called a cordial labeling if v f (0) v f (1) 1 and e f (0) e f (1) 1. A graph which admits cordial labeling is called a cordial graph. The cordial labeling technique was introduced by Cahit[2] in which he investigated several results on this concept. After this many labeling techniques are also introduced with minor changes in cordial labeling. The product cordial labeling, total product cordial labeling and prime cordial labeling are some of them. The present work is focused on divisor cordial labeling. Definition 1.5. Let G = (V (G), E(G)) be a simple graph and f : V (G) {1, 2,..., V (G) } be a bijection. For each edge e = uv, assign the label 1 if f (u) f(v) or f(v) f (u) and the label 0 otherwise. The function f is called a divisor cordial labeling if e f (0) e f (1) 1. A graph which admits divisor cordial labeling is called a divisor cordial graph. The divisor cordial labeling technique was introduced by Varatharajan et al.[7] and they proved the following results: The star graph K 1,n is divisor cordial. The complete bipartite graph K 2,n is divisor cordial. The complete bipartite graph K 3,n is divisor cordial. S(K 1,n ), the subdivision of the star K 1,n is divisor cordial. Vaidya and Shah [6] proved that S (B n,n ) is a divisor cordial graph. DS(B n,n ) is a divisor cordial graph. D 2 (B n,n ) is a divisor cordial graph. Restricted Bn,n 2 is a divisor cordial graph. Definition 1.6. Bistar B m,n is the graph obtained by joining the center(apex) vertices of K 1,m and K 1,n by an edge.

3 Divisor Cordial Labeling in the Context of Graph Operations on Bistar 2607 Definition 1.7. For a graph G the splitting graph S (G) of a graph G is obtained by adding a new vertex v corresponding to each vertex v of G such that N(v) = N(v ). Definition 1.8. [5] Let G = (V (G), E(G)) be a graph with V = S 1 S 2 S t T where each S i is a set of vertices having at least two vertices of the same degree and ( t T = V S i ). The degree splitting graph of G denoted by DS(G) is obtained from i=1 G by adding vertices w 1,w 2,w 3,...,w t and joining to each vertex of S i for 1 i t. Definition 1.9. The shadow graph D 2 (G) of a connected graph G is constructed by taking two copies of G say G and G". Join each vertex u in G to the neighbours of the corresponding vertex u"ing". Definition For a simple connected graph G the square of graph G is denoted by G 2 and defined as the graph with the same vertex set as of G and two vertices are adjacent in G 2 if they are at a distance 1 or 2 apart in G. We note that the restricted square graph of bistar B 2 m,n is the graph obtained from B m,n by joining all the pendant vertices of the K 1,m with the apex vertex of K 1,n and all the pendant vertices of the K 1,n with the apex vertex of K 1,m. Definition Let G = (V (G), E(G)) be a graph. Let e = uv be an edge of G and w is not a vertex of G. The edge e is sub divided when it is replaced by the edges e = uw and e = wv. Definition Let G = (V (G), E(G)) be a graph. If every edge of graph G is subdivided, then the resulting graph is called barycentric subdivision of graph G. In other words barycentric subdivision is the graph obtained by inserting a vertex of degree 2 into every edge of original graph. The barycentric subdivision of any graph G is denoted by S(G). Definition If G is graph of order n, the corona of G with another graph H, G H is the graph obtained by taking one copy of G and n copies of H and joining the i th vertex of G with an edge to every vertex in the i th copy of H. 2. Main Results Theorem 2.1. The bistar B m,n is a divisor cordial graph. Proof. Let G = B m,n be the bistar with vertex set V (G) ={u 0,v 0,u i,v j : 1 i m, 1 j n}, where u i and v j are pendant vertices. We note that V (G) =m + n + 2 and E(G) =m + n + 1. Without loss of generality we can assume that m n because B m,n and B n,m are isomorphic graphs. Define vertex labeling f : V (G) {1, 2,..., m + n + 2} as follows: f(u 0 ) = 1, f(v 0 ) = 2.

4 2608 M. I. Bosmia and K. K. Kanani f(u i ) = 2i + 2; 1 i m. 2j + 1; f(v j ) = 2m ( j m + n ) ; 2 m + n 1 j 2 m + n <j n 2 m + n + 1 In view of the above defined labeling pattern we have e f (0) = and e f (1) = 2 m + n + 1. Thus, e f (0) e f (1) 1. Hence, the bistar B m,n is a divisor cordial 2 graph. Illustration 2.2. Divisor cordial labeling of the graph B 4,8 is shown in the Figure 1. Figure 1: Divisor cordial labeling of B 4,8. Illustration 2.3. Divisor cordial labeling of the graph B 3,9 is shown in the Figure 2. Figure 2: Divisor cordial labeling of B 3,9.

5 Divisor Cordial Labeling in the Context of Graph Operations on Bistar 2609 Illustration 2.4. Divisor cordial labeling of the graph B 3,8 is shown in the Figure 3. Figure 3: Divisor cordial labeling of B 3,8. Illustration 2.5. Divisor cordial labeling of the graph B 5,5 is shown in the Figure 4. Figure 4: Divisor cordial labeling of B 5,5. Theorem 2.6. S (B m,n ) is a divisor cordial graph. Proof. Let B m,n be the bistar with vertex set V(B m,n ) ={u 0,v 0,u i,v j : 1 i m, 1 j n}, where u i and v j are pendant vertices. Let u 0,v 0,u i,v j be the newly added vertices in order to obtain G = S (B m,n ), where 1 i m and 1 j n. We note that V (G) =2m + 2n + 4 and E(G) =3m + 3n + 3. Without loss of generality we can assume that m n because S (B m,n ) and S (B n,m ) are isomorphic graphs. Define vertex labeling f : V (G) {1, 2,...,2m + 2n + 4} as follows: f(u 0 ) = 1, f(u 0 ) = 2m + 2n + 3. f(v 0 ) = 2, f(v 0 ) = 4. f(u i ) = 2n i; 1 i m. m + n f(u i (n ) ) = i; 1 i m. 2 m + n 4j + 2; 1 j f(v j ) = ( 2 m + n ) m + n 4 j + 4; <j n 2 2

6 2610 M. I. Bosmia and K. K. Kanani f(v j ) = 2j + 1; 1 j n. 3m + 3n + 3 In view of the above defined labeling pattern we have e f (0) = and 2 3m + 3n + 3 e f (1) =. Thus, e f (0) e f (1) 1. Hence, S (B m,n ) is a divisor 2 cordial graph. Illustration 2.7. Divisor cordial labeling of the graph S (B 3,7 ) is shown in the Figure 5. Figure 5: Divisor cordial labeling of S (B 3,7 ). Illustration 2.8. Divisor cordial labeling of the graph S (B 1,8 ) is shown in the Figure 6. Figure 6: Divisor cordial labeling of S (B 1,8 ).

7 Divisor Cordial Labeling in the Context of Graph Operations on Bistar 2611 Illustration 2.9. Divisor cordial labeling of the graph S (B 5,5 ) is shown in the Figure 7. Figure 7: Divisor cordial labeling of S (B 5,5 ). Theorem DS(B m,n ) is a divisor cordial graph. Proof. Let B m,n be the bistar with vertex set V(B m,n ) ={u 0,v 0,u i,v j : 1 i m, 1 j n}, where u i and v j are pendant vertices. Here, V(B m,n ) = V 1 V 2, where V 1 ={u i,v j : 1 i m, 1 j n} and V 2 ={u 0,v 0 }. Without loss of generality we can assume that m n because DS(B m,n ) and DS(B n,m ) are isomorphic graphs. Now in order to obtain G = DS(B m,n ) from B m,n, we consider following two cases. Case 1: m = n. We add w 1,w 2 corresponding to V 1,V 2. Then V (G) =2n+4 and E(G) = E(B n,n ) {u 0 w 2,v 0 w 2,u i w 1,v i w 1 : 1 i n}. So, E(G) =4n + 3. Define vertex labeling f : V (G) {1, 2,...,2n + 4} as follows. Let p be the largest prime number such that p<2n + 4. f(u 0 ) = 2, f(v 0 ) = 1. f(w 1 ) = p, f(w 2 ) = 3. f(u i ) = 2i + 2; 1 i n. Label the remaining vertices v 1,v 2,...,v n from the set {5, 7, 9,...,2n+3, 2n+4} {p}. In view of the above defined labeling pattern we have e f (0) = 2n + 1, e f (1) = 2n + 2. Thus, e f (0) e f (1) 1.

8 2612 M. I. Bosmia and K. K. Kanani Case 2: m<n. We add w 1 to V 1. Then V (G) =m + n + 3 and E(G) = E(B m,n ) {u i w 1,v j w 1 : 1 i m, 1 j n}. So, E(G) =2m + 2n + 1. Define vertex labeling f : V (G) {1, 2,..., m + n + 3} as follows: Let q be the largest prime number such that q m + n + 3. f(u 0 ) = 2, f(v 0 ) = 1. f(w 1 ) = q. f(u i ) = 2i + 2; 1 i m. Label the remaining vertices v 1,v 2,...,v n from the set {3, 5, 7,...,2m + 3, 2m + 4, 2m + 5,...,m+ n + 3} {q}. In view of the above defined labeling pattern we have e f (0) = m + n, e f (1) = m + n + 1. Thus, e f (0) e f (1) 1. Hence, DS(B m,n ) is a divisor cordial graph. Illustration Divisor cordial labeling of the graph DS(B 4,4 ) is shown in the Figure 8. Figure 8: Divisor cordial labeling of DS(B 4,4 ).

9 Divisor Cordial Labeling in the Context of Graph Operations on Bistar 2613 Illustration Divisor cordial labeling of the graph DS(B 3,6 ) is shown in the Figure 9. Figure 9: Divisor cordial labeling of DS(B 3,6 ). Theorem D 2 (B m,n ) is a divisor cordial graph. Proof. Let G and G" be two copies of bistar B m,n. Let V(G ) ={u 0,v 0,u i,v j : 1 i m, 1 j n} and V(G") ={u" 0,v" 0,u" i,v" j : 1 i m, 1 j n}. Let G = D 2 (B m,n ). We note that V (G) =2m + 2n + 4 and E(G) =4m + 4n + 4. Without loss of generality we can assume that m n because D 2 (B m,n ) and D 2 (B n,m ) are isomorphic graphs. Define vertex labeling f : V (G) {1, 2,...,2m + 2n + 4} as follows: Let p be the largest prime number and q be the second largest prime number such that q<p<2m + 2n + 4. f(u 0 ) = 2,f(u" 0) = q. f(v 0 ) = 1,f(v" 0) = p. f(u i ) = 2i + 2; 1 i m. f(u" i ) = 2m i; 1 i m. Label the remaining vertices v 1,v 2,...,v n,v" 1,v" 2,...,v" n from the set {3, 5, 7,..., 4m + 3, 4m + 4, 4m + 5,..., 2m + 2n + 4} {p, q}. In view of the above defined labeling pattern we have e f (0) = e f (1) = 2m + 2n + 2. Thus, e f (0) e f (1) 1. Hence, D 2 (B m,n ) is a divisor cordial graph.

10 2614 M. I. Bosmia and K. K. Kanani Illustration Divisor cordial labeling of the graph D 2 (B 4,8 ) is shown in the Figure 10. Figure 10: Divisor cordial labeling of D 2 (B 4,8 ). Theorem Restricted Bm,n 2 is a divisor cordial graph. Proof. Let B m,n be the bistar with vertex set V(B m,n ) ={u 0,v 0,u i,v j : 1 i m, 1 j n}, where u i and v j are pendant vertices. Let G be the restricted Bm,n 2 graph with V (G) = V(B m,n ) and E(G) = E(B m,n ) {v 0 u i,u 0 v j : 1 i m, 1 j n}. We note V (G) = m + n + 2 and E(G) = 2m + 2n + 1. Without loss of generality we can assume that m n because restricted Bm,n 2 and restricted B2 n,m are isomorphic graphs. Define vertex labeling f : V (G) {1, 2,..., m + n + 2} as follows: Let p be the largest prime number such that p m + n + 2. f(u 0 ) = p, f(v 0 ) = 1. f(u i ) = 2i; 1 i m. Label the remaining vertices v 1,v 2,...,v n from the set {3, 5, 7,...,2m + 1, 2m + 2, 2m + 3,...,m+ n + 2} {p}. In view of the above defined labeling pattern we have e f (0) = m + n and e f (1) = m + n + 1. Thus, e f (0) e f (1) 1. Hence, restricted is a divisor cordial graph. B 2 m,n

11 Divisor Cordial Labeling in the Context of Graph Operations on Bistar 2615 Illustration Divisor cordial labeling of the graph B2,6 2 is shown in the Figure 11. Figure 11: Divisor cordial labeling of B 2 2,6. Illustration Divisor cordial labeling of the graph B1,8 2 is shown in the Figure 12. Figure 12: Divisor cordial labeling of B 2 1,8. Theorem The barycentric subdivision S(B m,n ) of the bistar B m,n is a divisor cordial graph.

12 2616 M. I. Bosmia and K. K. Kanani Proof. Let B m,n be the bistar with vertex set V(B m,n ) ={u 0,v 0,u i,v j : 1 i m, 1 j n}, where u i and v j are pendant vertices and edge set E(B m,n ) ={u 0 v 0,u 0 u i,v 0 v j : 1 i m, 1 j n}. Let w 0,w 1,w 2,...,w m,w 1,w 2,...,w n be the newly added vertices to obtain G = S(B m,n ). Where w 0 is added between u 0 and v 0, w i is added between u 0 and u i for 1 i m and w j is added between v 0 and v j for 1 j n. We note that V (G) =2m + 2n + 3 and E(G) =2m + 2n + 2. Define vertex labeling f : V (G) {1, 2,...,2m + 2n + 3} as follows: f(u 0 ) = 2, f(v 0 ) = 1. f(w 0 ) = 3. f(u i ) = 2i + 3; 1 i m. f(w i ) = 2i + 2; 1 i m. f(v j ) = 2m j; 1 j n. f(w j ) = 2m j; 1 j n. In view of the above defined labeling pattern we have e f (0) = e f (1) = m+n+1. Thus, e f (0) e f (1) 1. Hence, the barycentric subdivision S(B m,n ) of the bistar B m,n is a divisor cordial graph. Illustration Divisor cordial labeling of the graph S(B 3,7 ) is shown in the Figure 13. Figure 13: Divisor cordial labeling of S(B 3,7 ). Theorem B m,n K 1 is a divisor cordial graph. Proof. Let B m,n be the bistar with vertex set V(B m,n ) ={u 0,v 0,u i,v j : 1 i m, 1 j n}, where u i and v j are pendant vertices. Let u 0,v 0,u 1,u 2,...,u m,v 1,v 2,...,v n be the newly added vertices to obtain the graph G = B m,n K 1. We note that V (G) = V(B m,n ) {u 0,v 0,u i,v j : 1 i m, 1 j n} and E(G) = E(B m,n) {u 0 u 0,v 0v 0,u iu i,v jv j : 1 i m, 1 j n}. Hence, V (G) =2m + 2n + 4 and E(G) =2m + 2n + 3.

13 Divisor Cordial Labeling in the Context of Graph Operations on Bistar 2617 Define vertex labeling f : V (G) {1, 2,...,2m + 2n + 4} as follows: f(u 0 ) = 1,f(u 0 ) = 2m + 2n + 4. f(v 0 ) = 2,f(v 0 ) = 3. f(u i ) = 2i + 2; 1 i m. f(u i ) = 2i + 1; 1 i m. f(v j ) = 2m j; 1 j n. f(v j ) = 2m j; 1 j n. In view of the above defined labeling pattern we have e f (0) = m + n + 1 and e f (1) = m + n + 2. Thus, e f (0) e f (1) 1. Hence, B m,n K 1 is a divisor cordial graph. Illustration Divisor cordial labeling of the graph B 5,7 K 1 is shown in the Figure 14. Figure 14: Divisor cordial labeling of B 5,7 K Concluding Remark To explore some new divisor cordial graphs is an open problem. References [1] D. M. Burton, Elementary Number Theory, Brown Publishers, Second Edition, (1990). [2] I. Cahit, Cordial Graphs, A weaker version of graceful and harmonious graphs, Ars Combinatoria, 23(1987),

14 2618 M. I. Bosmia and K. K. Kanani [3] J. A. Gallian, A dynamic Survey of Graph labeling, The Electronics Journal of Combinatorics, 18(2015), # D56. [4] J. Gross and J. Yellen, Handbook of Graph Theory, CRC Press, (2004). [5] P. Selvaraju, P. Balaganesan, J. Renuka and V. Balaj, Degree spliting graph on graceful, felicitous and elegant labeling, International journal of Mathematical combinatorics, 2(2012), [6] S. K. Vaidya, N. H. Shah, Some star and bistar related divisor cordial graphs, Annals of Pure and Applied Mathematics, 3(1) (2013), [7] R.Varatharajan, S.Navanaeethakrishnan, K.Nagarajan, Divisor cordial graphs, International Journal of Mathematical Combinatorics, 4(2011),