P. Jeyanthi and N. Angel Benseera


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1 Opuscula Math. 34, no. 1 (014), Opuscula Mathematica A TOTALLY MAGIC CORDIAL LABELING OF ONEPOINT 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 onepoint union of ncopies of cycles, complete graphs and wheels. Keywords: totally magic cordial labeling, onepoint 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 onepoint union of n copies of graphs is given in [6]. Kotzig and Rosa introduced the concept of edgemagic total labeling in [5]. A bijection f : V (G) E(G) 1,, 3,..., p + q} is called an edgemagic 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 edgemagic 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 onepoint union of n copies of G and is denoted by G (n). In this paper, we establish the TMC labeling of a onepoint union of ncopies of cycles, complete graphs and wheels.. MAIN RESULTS In this section, we present sufficient conditions for a onepoint union of n copies of a rooted graph to be TMC and also obtain conditions under which a onepoint union of n copies of graphs such as a cycle, complete graph and wheel are TMC graphs. We relate the TMC labeling of a onepoint 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 onepoint 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 onepoint 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. ONEPOINT 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 onepoint 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 ONEPOINT 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 subcomplete graph in which all the vertices and edges are labeled with 1, K r is the subcomplete 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 m1 (m 1) 3 m 3 (m ) 4 3 m3 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 onepoint 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 onepoint 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 onepoint 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 ONEPOINT 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 onepoint 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 onepoint 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, AddisonWesley 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] SzeChin Shee, YongSong Ho, The cordiality of onepoint union of n copies of a graph, Discrete Math. 117 (1993), P. Jeyanthi 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.
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