P. Jeyanthi and N. Angel Benseera

Transcription

1 Opuscula Math. 34, no. 1 (014), Opuscula Mathematica A TOTALLY MAGIC CORDIAL LABELING OF ONE-POINT UNION OF n COPIES OF A GRAPH P. Jeyanthi and N. Angel Benseera Communicated by Dalibor Fronček Abstract. A graph G is said to have a totally magic cordial (TMC) labeling with constant C if there exists a mapping f : V (G) E(G) 0, 1} such that f(a)+f(b)+f(ab) C(mod ) for all ab E(G) and n f (0) n f (1) 1, where n f (i) (i = 0, 1) is the sum of the number of vertices and edges with label i. In this paper, we establish the totally magic cordial labeling of one-point union of n-copies of cycles, complete graphs and wheels. Keywords: totally magic cordial labeling, one-point union of graphs. Mathematics Subject Classification: 05C INTRODUCTION All graphs considered here are finite, simple and undirected. The set of vertices and edges of a graph G is denoted by V (G) and E(G) respectively. Let p = V (G) and q = E(G). A general reference for graph theoretic ideas can be seen in [3]. The concept of cordial labeling was introduced by Cahit [1]. A binary vertex labeling f : V (G) 0, 1} induces an edge labeling f : E(G) 0, 1} defined by f (uv) = f(u) f(v). Such labeling is called cordial if the conditions v f (0) v f (1) 1 and e f (0) e f (1) 1 are satisfied, where v f (i) and e f (i) (i = 0, 1) are the number of vertices and edges with label i respectively. A graph is called cordial if it admits a cordial labeling. The cordiality of a one-point union of n copies of graphs is given in [6]. Kotzig and Rosa introduced the concept of edge-magic total labeling in [5]. A bijection f : V (G) E(G) 1,, 3,..., p + q} is called an edge-magic total labeling of G if f(x) + f(xy) + f(y) is constant (called the magic constant of f) for every edge xy of G. The graph that admits this labeling is called an edge-magic total graph. The notion of totally magic cordial (TMC) labeling was due to Cahit [] as a modification of edge magic total labeling and cordial labeling. A graph G is said to have TMC labeling with constant C if there exists a mapping f : V (G) E(G) 0, 1} c AGH University of Science and Technology Press, Krakow

2 116 P. Jeyanthi and N. Angel Benseera such that f(a) + f(b) + f(ab) C(mod ) for all ab E(G) and n f (0) n f (1) 1, where n f (i) (i = 0, 1) is the sum of the number of vertices and edges with label i. A rooted graph is a graph in which one vertex is named in a special way so as to distinguish it from other nodes. The special node is called the root of the graph. Let G be a rooted graph. The graph obtained by identifying the roots of n copies of G is called the one-point union of n copies of G and is denoted by G (n). In this paper, we establish the TMC labeling of a one-point union of n-copies of cycles, complete graphs and wheels.. MAIN RESULTS In this section, we present sufficient conditions for a one-point union of n copies of a rooted graph to be TMC and also obtain conditions under which a one-point union of n copies of graphs such as a cycle, complete graph and wheel are TMC graphs. We relate the TMC labeling of a one-point union of n copies of a rooted graph to the solution of a system which involves an equation and an inequality. Theorem.1. Let G be a graph rooted at a vertex u and for i = 1,,..., k, f i : V (G) E(G) 0, 1} be such that f i (a) + f i (b) + f i (ab) C(mod ) for all ab E(G) and f i (u) = 0. Let n fi (0) = α i, n fi (1) = β i for i = 1,,..., k. Then the one-point union G (n) of n copies of G is TMC if the system (.1) has a nonnegative integral solution for the x i s: k k k (α i 1)x i β i x i and x i = n. (.1) i=1 i=1 Proof. Suppose x i = δ i, i = 1,,..., k, is a nonnegative integral solution of system (.1). Then we label the δ i copies of G in G (n) with f i (i = 1,,..., k). As each of these copies has the property f i (a) + f i (b) + f i (ab) C(mod ) and f i (u) = 0 for all i = 1,,..., k, G (n) is TMC. Corollary.. Let G be a graph rooted at a vertex u and f be a labeling such that f(a)+f(b)+f(ab) C(mod ) for all ab E(G) and f(u) = 0. If n f (0) = n f (1)+1, then G (n) is TMC for all n 1. Example.3. One point union of a path is TMC. Corollary.4. Let G be a graph rooted at u. Let f i, i = 1,, 3 be labelings of G such that f i (a) + f i (b) + f i (ab) C(mod ) for all ab E(G), f i (u) = 0 and γ i = α i β i. 1. If γ 1 = and γ =, then G (n) is TMC for all n 1(mod 4).. If either a) γ 1 = 1 and γ = 3, or b) γ 1 = 4, γ = and γ 3 = 4, or c) γ 1 = 3, γ = 3 and γ 3 = 5, then G (n) is TMC for all n If γ 1 = 0 and γ = 4, then G (n) is TMC for all n 3(mod 4). i=1

3 A totally magic cordial labeling of one-point union of n copies of a graph 117 Proof. (1) The system (.1) in Theorem.1 becomes 3x 1 + x + 1 1, x 1 +x = n. When n = 4t, x 1 = t and x = 3t is the solution. When n = 4t + 1, the system has no solution. When n = 4t +, x 1 = t + 1 and x = 3t + 1 is the solution. When n = 4t + 3, x 1 = t + 1 and x = 3t + is the solution. Hence, by Theorem.1, G (n) is TMC for all n 1(mod 4). (a). The system (.1) in Theorem.1 becomes x 1 + x + 1 1, x 1 +x = n. When n = t, x 1 = t and x = t is the solution. When n = t + 1, x 1 = t + 1 and x = t is the solution. Hence, by Theorem.1, G (n) is TMC for all n 1. The other parts can similarly be proved. 3. ONE-POINT UNION OF CYCLES Let C m be a cycle of order m. Let and V (C m ) = v i 1 i m} E(C m ) = v i v i+1 1 i < m} v m v 1 }. We consider C m as a rooted graph with the vertex v 1 as its root. Theorem 3.1. Let C m (n) be the one-point union of n copies of a cycle C m. Then C m (n) is TMC for all m 3 and n 1. Proof. Define the labelings f 1 and f from V (C m ) E(C m ) into 0, 1} as follows: f 1 (v i ) = 0 for 1 i m, f 1 (v i v i+1 ) = 1 for 1 i < m, f 1 (v m v 1 ) = 1, 1 i m and 1 if i = m, 1 if 1 i < m 1, f (v i ) = f (v i v i+1 ) = 0 if i m, 0 if i = m 1, and f (v m v 1 ) = 0. Then α 1 = m, β 1 = m, α = m + 1 and β = m 1. Thus system (.1) in Theorem.1 becomes x 1 + x + 1 1, x 1 + x = n. When n = t, x 1 = t and x = t is the solution. When n = t + 1, x 1 = t + 1 and x = t is the solution. Hence, by Theorem.1, C m (n) is TMC for all m 3 and n ONE-POINT UNION OF COMPLETE GRAPHS Let K m be a complete graph of order m. Let and V (K m ) = v i 1 i m} E(K m ) = v i v j i j, 1 i m, 1 j m}. We consider K m as a rooted graph with the vertex v 1 as its root. Let f : V (K m ) E(K m ) 0, 1} be a TMC labeling of K m. Without loss of generality, assume C = 1.

4 118 P. Jeyanthi and N. Angel Benseera Then for any edge e = uv E(K m ), we have either f(e) = f(u) = f(v) = 1 or f(e) = f(u) = 0 and f(v) = 1 or f(e) = f(v) = 0 and f(u) = 1 or f(u) = f(v) = 0 and f(e) = 1. Thus, under the labeling f, the graph K m can be decomposed as K m = K p K r K p,r where K p is the sub-complete graph in which all the vertices and edges are labeled with 1, K r is the sub-complete graph in which all the vertices are labeled with 0 and edges are labeled with 1 and K p,r is the complete bipartite subgraph of K m with the bipartition V (K p ) V (K r ) and its edges are labeled with 0. Then we find n f (0) = r + pr and n f (1) = p +r +p r. Table 1. Possible values of α i and β i for distinct labelings of K m i p r α i β i 1 0 m m 1 m-1 (m 1) 3 m- 3 (m ) 4 3 m-3 4 (m 3) m+1 m 1 m+1 m 1 m+1 m m m 3m+4 m 5m+1 m 7m+4 [ ( m 1 ) +( m+1 ) + m 1 + m+1 ] Table 1 gives the possible values of α i and β i for distinct labelings f i of K m such that f i (a) + f i (b) + f i (ab) 1(mod ) for all ab E(K m ). Theorem 4.1. Let K m (n) be the one-point union of n copies of a complete graph K m. If m 1 has an integer value, then K m (n) is TMC for m 1, (mod 4). Proof. Let f : V (K m ) E(K m ) 0, 1} be a TMC labeling of K m. Under the labeling f, the graph K m can be decomposed as K m = K p K r K p,r. Then we have, n f (0) = r + pr and n f (1) = p +r +p r. By Corollary., K m (n) is TMC if n f (0) = n f (1) + 1. Whenever, [ n f (0) = n f (1) + 1, p + p(1 r) + r 3r + = 0. This implies that r = 1 ] (m + 1) ± m 1 as p = m r. Also, nf (0) = n f (1) + 1 is possible only when m 1, (mod 4). Therefore, K m (n) is TMC for m 1, (mod 4), if m 1 has an integer value Theorem 4. ([4]). Let G be an odd graph with p + q (mod 4). Then G is not TMC. Theorem 4.3. Let K (n) m be the one-point union of n copies of a complete graph K m. (i) If m 0(mod 8), then K m (n) is not TMC for n 3(mod 4). (ii) If m 4(mod 8), then K m (n) is not TMC for n 1(mod 4). Proof. Clearly, p = V (Km) n = n(m 1) + 1 and q = E(Km) n = nm(m 1) so that p + q = n(m 1)(m+) + 1. Part (i) Assume m = 8k and n = 4l + 3. Since the degree of every vertex is odd and

5 A totally magic cordial labeling of one-point union of n copies of a graph 119 p + q (mod 4), it follows from Theorem 4. that K m (n) Part (ii) can similarly be proved. Theorem 4.4. K (n) 4 is TMC if and only if n 1(mod 4). is not TMC. Proof. Necessity follows from Theorem 4.3 and for sufficiency we define the labelings f 1 and f as follows: f 1 (v i ) = 0 for 1 i 4, f 1 (v i v j ) = 1 for 1 i, j 4 and under the labeling f decompose K 4 as K 1 K 3 K 1,3. From Table 1, we observe that α 1 = 4, β 1 = 6, α = 6 and β = 4. Therefore, by Corollary.4 (1), K (n) 4 is TMC if n 1(mod 4). Theorem 4.5. K (n) 5 is TMC for all n 1. Proof. Define f : V (K (n) 5 ) E(K (n) 5 ) 0, 1} as follows: 0 if i 5, f(v i ) = 1 if i = 5 and f(v i v j ) = 1 if 1 i, j 4, 0 if i = 5 or j = 5. Clearly, α = β + 1 = 8. Therefore, by Corollary., K (n) 5 is TMC for all n 1. Theorem 4.6. K (n) 6 is TMC for all n 1. Proof. Let f 1 and f be the labelings from V (K (n) 6 ) E(K (n) 6 ) into 0, 1}. Then, under the labelings f 1 and f the graph K 6 can be decomposed as K 1 K 5 K 1,5 and K K 4 K,4 respectively. Clearly, α 1 = 10, β 1 = 11, α = 1 and β = 9. Hence, by Corollary.4 (a), K (n) 6 is TMC for all n 1. Theorem 4.7. K (n) 7 is TMC for all n 1. Proof. Let f 1, f and f 3 be the labelings from V (K (n) 7 ) E(K (n) 7 ) into 0, 1}. Then under the labelings f 1, f and f 3 the graph K 7 can be decomposed as K 3 K 4 K 3,4, K 4 K 3 K 4,3 and K 5 K K 5, respectively. We observe that α 1 = 16, β 1 = 1, α = 15, β = 13, α 3 = 1 and β 3 = 16. Hence, by Corollary.4 (b), K (n) 7 is TMC for all n 1. Theorem 4.8. K (n) 8 is TMC if and only if n 3(mod 4). Proof. Necessity follows from Theorem 4.3 and for sufficiency we define the labelings f 1 and f as follows: under the labelings f 1 and f the graph K 8 can be decomposed as K K 6 K,6 and K 3 K 5 K 3,5 respectively. Clearly, α 1 = 18, β 1 = 18, α = 0 and β = 16. Hence, by Corollary.4 (3), K (n) 8 is TMC if n 3(mod 4). Theorem 4.9. K (n) 9 is TMC for all n 1.

6 10 P. Jeyanthi and N. Angel Benseera Proof. Under the labelings f 1, f and f 3 the graph K 9 can be decomposed as K K 7 K,7, K 3 K 6 K 3,6 and K 4 K 5 K 4,5 respectively. We observe that α 1 = 1, β 1 = 4, α = 4, β = 1, α 3 = 5 and β 3 = 0. Therefore, by Corollary.4 (c), the graph K (n) 9 is TMC for all n ONE-POINT UNION OF WHEELS A wheel W m is obtained by joining the vertices v 1, v,..., v m of a cycle C m to an extra vertex v called the centre. We consider W m as a rooted graph with v as its root. Theorem 5.1. Let W (n) m be the one-point union of n copies of a wheel W m. (i) If m 0(mod 4), then W m (n) is TMC for all n 1. (ii) If m 1(mod 4), then W m (n) is TMC for n 3(mod 4). (iii) If m (mod 4), then W m (n) is TMC for all n 1. (iv) If m 3(mod 4), then W m (n) is TMC for n 1(mod 4). Proof. Define the labelings f 1,f,f 3,f 4 and f 5 as follows: f j (v) = 0 for j = 1,, 3, 4, 5. f 1 (v m v 1 ) = 0, f 1 (v i ) = 1 if i 0(mod 4), 0 if i 0(mod 4), f 1 (v i v i+1 ) = 1 if i 1, (mod 4), 0 if i 0, 3(mod 4) and f 1 (vv i ) = 1 if i 0(mod 4), 0 if i 0(mod 4). f (v i ) = f (v i v i+1 ) = 1, f (vv i ) = 0 for i = 1,,..., m and f (v m v 1 ) = 1. f 3 (v i ) = f 1 (v i ), f 3 (v i v i+1 ) = f 1 (v i v i+1 ), f 3 (vv i ) = f 1 (vv i ) for i = 1,,..., m and f 3 (v m v 1 ) = 1. f 4 (v 1 ) = 1, f 4 (v 1 v ) = f 4 (v m v 1 ) = 0, f 4 (v i ) = f 3 (v i ), f 4 (v i v i+1 ) = f 3 (v i v i+1 ), f 4 (vv i ) = f 3 (vv i ) for i =, 3,..., m and f 4 (vv 1 ) = 0. f 5 (v i ) = 1 if i 1(mod ), 0 if i 0(mod ), f 5 (vv i ) = 0 if i 1(mod ), 1 if i 0(mod ), f 5 (v i v i+1 ) = f 5 (v m v 1 ) = 0. Case 1. m 0 (mod 4). If we consider the labeling f 1 we have, n f1 (0) = n f1 (1) + 1. Then, by Corollary., W m (n) is TMC for all n 1. Case. m 1 (mod 4). If we consider the labelings f, f 3 and f 4. We have α = 3m+1, β = 3m+1, α 3 = 3m+5, β 3, α 4 = m + 1, β 4 = m. Then, system (.1) in Theorem.1 becomes x + 3x 3 (m + 1)x , x + x 3 + x 4 = n. When n = 4t, x = 3t, x 3 = t, x 4 = 0 is a solution. When n = 4t + 1, x = 3t + 1, x 3 = t, x 4 = 0 is a solution. When = 3m 3 n = 4t +, x = 3t +, x 3 = t, x 4 = 0 is a solution. When n = 4t + 3, the system has no solution. Hence, by Theorem.1, W (n) m is TMC if n 3(mod 4).

7 A totally magic cordial labeling of one-point union of n copies of a graph 11 Case 3. m (mod 4). If we consider the labelings f, f 3, f 4 and f 5, we have α = m+1, β = m, α 3 = 3m, β 3 = 3m+, α 4 = 3m+4, β 4 = 3m, α 5 = m + 1, β 5 = m. Thus, system (.1) in Theorem.1 becomes mx x 3 + x 4 + mx , x + x 3 + x 4 + x 5 = n. When n = 4t, x = x 3 = x 4 = x 5 = t is a solution. When n = 4t + 1, x = t, x 3 = t + 1, x 4 = t, x 5 = t is a solution. When n = 4t +, x = t + 1, x = t, x 4 = t, x 5 = t + 1 is a solution. When n = 4t + 3, x = t + 1, x 3 = t + 1, x 4 = t, x 5 = t + 1 is a solution. Hence, by Theorem.1, W (n) m is TMC for all n 1. Case 4. m 3 (mod 4). If we consider the labelings f 3 and f 4. We have α 3 = 3m 1, β 3 = 3m+3, α 4 = 3m+3 and β 4 = 3m 1. Therefore, system (.1) in Theorem.1 becomes, 3x 3 + x , x 3 + x 4 = n. When n = 4t, x 3 = t, x 4 = 3t is a solution. When n = 4t + 1,the system has no solution. When n = 4t +, x 3 = t + 1, x 4 = 3t + 1 is a solution. When n = 4t + 3, x 3 = t + 1, x 4 = 3t + is a solution. Hence, by Theorem.1, W m (n) is TMC if n 1(mod 4). Acknowledgments The authors sincerely thank the referee for the valuable suggestions which were used in this paper. REFERENCES [1] I. Cahit, Cordial graphs: A weaker version of graceful and harmonious graphs, Ars Combin. 3 (1987), [] I. Cahit, Some totally modular cordial graphs, Discuss. Math. Graph Theory (00), [3] F. Harary, Graph Theory, Addison-Wesley Publishing Co., [4] P. Jeyanthi, N. Angel Benseera, Totally magic cordial labeling for some graphs (preprint). [5] A. Kotzig, A. Rosa, Magic valuations of finite graphs, Canad. Math. Bull. 13 (1970) 4, [6] Sze-Chin Shee, Yong-Song Ho, The cordiality of one-point union of n copies of a graph, Discrete Math. 117 (1993), P. Jeyanthi jeyajeyanthi@rediffmail.com Research Center Department of Mathematics Govindammal Aditanar College for Women Tiruchendur 68 15, India

8 1 P. Jeyanthi and N. Angel Benseera N. Angel Benseera Department of Mathematics Sri Meenakshi Government Arts College for Women (Autonomous) Madurai 65 00, India Received: June 10, 013. Revised: September 14, 013. Accepted: September 1, 013.