On Z 3 -Magic Labeling and Cayley Digraphs

Transcription

1 Int. J. Contemp. Math. Sciences, Vol. 5, 00, no. 48, On Z 3 -Magic Labeling and Cayley Digraphs J. Baskar Babujee and L. Shobana Department of Mathematics Anna University Chennai, Chennai , India baskarbabujee@yahoo.com, shobana 0@yahoo.com Abstract Let A be an abelian group. An A-magic of a graph G = (V, E) is a labeling f : E(G) A\{0} such that the sum of the labels of the edges incident with u V is a constant, where 0 is the identity element of the group A. In this paper we prove Z 3 -magic labeling for the class of even cycles, Bistar, ladder, biregular graphs and for a certain class of Cayley digraphs. Mathematics Subject Classification: 05C78 Keywords: Group-magic, A-magic, labeling, cayley digraphs Introduction Let G be a connected or multi graph, with no loops. For any additively abelian group A, let A = A {0}. A function f : E(G) A is called the labeling of G. Any such labeling induces a map f : V (G) A defined by f (v) = f(u,v), where (u,v) E(G). If there exists a labeling f whose induced map on V (G) is a constant map, we say that f is an A-magic labeling and that G is an A-magic graph. The original concept of an A-magic graph is due to Sedlacek [9, 0], who defined it to be a graph with real-valued edge labeling such that distinct edges have distinct nonnegative labels. Though different kinds of labeling were studied and many conjectures were made for different subclasses of graphs the labeling of well known graph namely Cayley graphs has not been investigated for Z 3 -magic labeling. The Cayley digraph of a group provides a method of visualizing the group and its properties. The properties such as commutativity and the multiplication table of a group can be recovered from a Cayley digraph. A directed graph or digraph is a finite set of points called vertices and a set of arrows called arcs connecting some of the vertices. The idea of representing a group

2 358 J. Baskar Babujee and L. Shobana in such a manner was originated by Cayley in 878. The Cayley graphs and Cayley digraphs are excellent models for interconnection networks [, ]. Many well-known interconnection networks are Cayley digraphs. For example, hypercube,butterfly and cube-connected cycle s networks are Cayley graphs [5]. The Cayley digraph for a given finite group G with S as a set of generators of G has two properties. The first property is that each element of G is a vertex of the Cayley digraph. The second property is for a and b in G, there is an arc from a to b if and only if as = b for some s in S. It is easy to verify whether a graph is Z -magic or not. It seems that finding a similar condition for G to be Z 3 -magic is much more difficult. The sum of the labels of the edges incident to a particular vertex is the same for all vertices. In this paper we prove that even cycles, complete graphs, Bistar, ladder, a class of biregular graphs and for a certain class of Cayley digraphs admits Z 3 -magic labeling. Theorem.. ([7] (Richard.M. Low and S-M. Lee)) Let G be Z 3 -magic, with p vertices and q edges. Let f [G,Z 3 ] induce the constant label x on the vertices of G, and E i denote the number of edges labeled i. Then, px q + E, (mod 3). Proof. With any Z 3 -magic labeling f of G, we can associate a multigraph G with G. G is formed by replacing every edge in G which was labeled, with two edges labeled. Note that G is a Z 3 -magic multi-graph with p vertices and E + E edges. In any (p,q)-graph, we have that deg(vi) = q. Since all of the edges in G are labeled, this implies that px E + E + E, (mod 3). Thus, px q + E, (mod 3). Theorem.. (W. C. Shiu, P. C. B. Lam, and P. K. Sun, Construction of group-magic graphs and some A-magic graphs with A of even order, Congr. Numer., 67 (004) 97-07) Let A be an abelian group of even order. A graph G is A-magic if degrees of its vertices are either all odd or all even. Proof. Let a be an element of A of order. We label all the edges of G by a. Then l (v) = a or 0 for all v V (G) as the degree of v is odd or even respectively. Z 3 -Magic Labeling Example.. The even cycle graph C n, n 0 mod is Z 3 -magic. The Figure illustrates Z 3 -magicness of C 6.

3 On Z 3 -magic labeling and Cayley digraphs 359 v 5 v 4 v 3 v 6 v v Figure : Algorithm.. Input: n, the number of vertices of cycle graph C n. Output: Construction of C n along with Z 3 -magic labeling. Step : V = {v,v,...,v n } Step : E = {v i v i+ : i n } {v n v } Step 3 : For i n{ if i mod f(v i v i+ ) = if i 0 mod and f(v n v ) =.. Theorem.3. The cycle graph C n admits Z 3 -magic labeling for n even. Proof. The above Algorithm. generates a mapping f : E Z 3 {0} to the cycle graph C n. We need to prove that the induced mapping f + : V Z 3 is constant. For any vertex v k { V and k n, f + + if k is odd (v k ) = v k v k+ + v k v k = + if k is even 0 mod 3. Also f + (v n ) = v n v + v n v n = + 0 mod 3 and f + (v ) = v v + v n v = + 0 mod 3. Hence f + (v k ) = 0 for k and C n : n even is Z 3 -magic. Algorithm.4. Input: n, the number of vertices of complete graph K n. Output: Z 3 -magic labeling for K n. Step : V = {v,v,...,v n } Step : E = {v i v j : i n, j n : i < j}

4 360 J. Baskar Babujee and L. Shobana Step 3 : f(v v n ) = f(v i v i+ ) = { i n otherwise Theorem.5. For n 4, the complete graph K n is Z 3 -magic. Proof. Consider the mapping f : E Z 3 {0} then we have to prove that the induced mapping f + : V Z 3 is constant. Since complete graphs are pairwise adjacent, we define f + (v k ) to be f + (v k ) = n j=,j k v kv j (n 4) mod 3. Case : When n = 4, 7, 0,...,(3k + ) then the Z 3 -magic labeling for K 4, K 7, K 0,..., K 3k+ is defined as follows. The induced map f + is given by f + (n 4) mod 3. When n = 3k +, then using Algorithm.4. f + (v k ) = (3k + ) 4 = 6k + 4 mod 3. Hence Z 3 -magic labeling for n = 3k + is constantly. Case : When n = 5, 8,,...,(3k + ), the induced mapping f + is given by, f + (v k ) = (3k + ) 4 = 6k mod 3. Therefore the Z 3 -magic labeling for n = 3k + is 0. Case 3: When n = 6, 9,,...,(3k + 3), the induced mapping f + is given by, f + (v k ) = (3k + 3) 4 = 6k mod 3. Therefore f + (v k ) is constantly for k. Hence the complete graph K n is Z 3 -magic. Theorem.6. The Bistar B 3k,3k k is Z 3 -magic.

5 On Z 3 -magic labeling and Cayley digraphs 36 Proof. V = {v,v,...,v n+ }, E = {v v n+ } {v v i : i n + } {v n+ v j : n + 3 j n + }, f(v v i ) = f(v n+ v j ) = for i n + and n + 3 j n + and f(v v n+ ) =. The induced map f + is given by f + (v k ) = (3k ) + = 6k + mod 3. Therefore f + (v k ) is constantly for k. Hence the Bistar B 3k,3k k is Z 3 -magic. Theorem.7. The Ladder L n, n 4 is Z 3 -magic. Proof. Case : (n even) Suppose n is even, n 4 then the edges are labeled in the following way: Step : V = {v,v,...,v n } Step : E = {v i v i+ : i n } {v j v j+ : n + j n } {v k v n+k : k n} { i mod Step 3 : f(v i v i+ ) = f(v n+i v n+i+ ) = i 0 mod { i n Step 4 : f(v i v n+i ) = i =,n The induced map f is defined by f + (v ) = f + (v n+ ) = f + (v n ) = f + (v n ) = + mod 3 and f + (v k ) = f + (v k+n ) = + +, k =,,...,n and k n mod 3 f + (v k+ ) = f + (v k+n+ ) = + +, k =,,...,n and k n mod 3 Therefore when n is even, then in ladder L n, f + (v k ) is constantly for k. Case : (n odd) When n is odd, the Z 3 -magic labeling for the ladder L n is defined as follows Step : V = {v,v,...,v n } Step : E = {v i v i+ : i n } {v j v j+ : n + j n } {v k v n+k : k n} Step 3 : f(v i v i+ ) = f(v n+i v n+i+ ) = when i n and f(v v )f(v n v n ) = f(v n+ v n+ ) = f(v n v n ) = Step 4 : f(v i v n+i ) =, 3 i n, f(v v n+ ) = f(v n v n ) = and

6 36 J. Baskar Babujee and L. Shobana f(v v n+ ) = f(v n v n ) =. The induced map f + is given by f + (v ) = f + (v n ) = f + (v n+ = f + (v n ) = + mod 3 f + (v i ) = + + mod 3,i =,n + = + + mod 3,i = n, n f + (v ) = f + (v i+n ) = + + mod 3,i = 3, 4,...,n Therefore f + (v ) is constantly for every i. Hence the ladder L n, n 4 is Z 3 -magic. Algorithm.8. Input: m,n, the number of vertices in the Biregular bipartite graph. Output: Construction of the Z 3 -magic labeling for the Biregular bipartite graph. Step : V = V V where V = {u,u,...,u m } and V = {v,v,...,v m } Step : E = E E E 3 E = {v k u k,v k u k+,k =,,...,m }, E = {v m u m,v m u,v m u } and E 3 = {v j u p,v j u p+ : p m,m + j m } Step 3 : f(v k u k,v j u q ) =,k =,,...,m ;m + j m, q m f(v k u k+,v j u q ) =,k =,,...,m ;m + j m, q m f(v m u,v m u m ) = and f(v m u m,v m u ) = u u u 3 v v v 3 v 4 v 5 v 6 Figure : The biregular bipartite graph K 3,6 Theorem.9. The biregular bipartite graph K m,n, m 3 and n = m is Z 3 -magic. Proof. For the (,4)-biregular bipartite graph K m,n, m 3 and n = m, deg(u i ) = 4 and deg(v j ) =. By the above Algorithm.8 we assign labels

7 On Z 3 -magic labeling and Cayley digraphs 363 for all the edges. We have to prove that the induced map f + : V Z 3 is constant. The induced map f + is given by f + (v k ) = v k u k + v k u k+ = + 0 mod 3,k =,,...,m for all edges v k u i E. f + (v m ) = v m u m + v m u = + 0 mod 3 and f + (v m ) = v m u m + v m u = + 0 mod 3 for all edges v k u j E. f + (v k ) = v k u i + v k u i+ = + 0 mod 3,k = m +,m +,...,m and i =,,...,m for all edges v k u l E 3. Hence the biregular bipartite graph K m,n, m 3 and n = m is Z 3 -magic. 3 Z 3 -Magic Labeling for Cayley Digraphs Definition 3.. Let G be a finite group, and let S be a generating subset of G. The Cayley digraph Cay(G; S) is the digraph whose vertices are the elements of G, and there is an edge from g to gs whenever g G and s S. If S = S, then there is an arc from g to gs if and only if there is an arc from gs to g. We present an Algorithm 3. to get Z 3 -magic labeling for Cayley digraph Cay(D n, (a,b)). Algorithm 3.. Input: The dihedral group Dn with the generating set (a,b). Step : Using Definition 3., construct the Cayley digraph Cay(D n, (a,b)) Step : Denote the vertex set of Cay(D n, (a,b)) as V = {v,v,v 3,...,v n } Step 3 : Denote the edge set of Cay(D n, (a,b)) as E(E a,e b ) = {e,e,e 3,...,e n } where E a = Set of all going arcs from v i generated by a E b = Set of all going arcs from v i generated by b Step 4 : Define f such that f(v i ) = i, { for i n ; i n/ Step 5 : Define g a such that g a (v i ) = ; (n/) + i n { ; i n/ Step 6 : Define g b such that g b (v i ) = ; (n/) + i n Output: Z 3 -magic labeling for Cayley digraph Cay(D n, (a,b)). Theorem 3.3. The Cayley digraph associated with dihedral group D n admits Z 3 -magic labeling. Proof. From the construction of the cayley digraph for the dihedral group D n, we have Cay(D n, (a,b)) has n vertices and n arcs. Let us denote the vertex set

8 364 J. Baskar Babujee and L. Shobana of Cay(D n, (a,b)) as V = {v,v,v 3,...,v n }. To prove Cay(D n, (a,b)) admits Z 3 -magic labeling we have to show that for any vertex v i, the sum of the labels of the outgoing arcs are constant. By construction of the Cayley digraph, we have each vertex has exactly two out going arcs out of which one is from the set E a and another is from the set E b, where E a and E b are defined in the above algorithm. Now define maps f, g a and g b as defined in steps 4, 5 and 6 of the above algorithm. Hence g a (v i ) + g b (v i ) = + = + = 0 mod 3. Therefore the Cayley digraph Cay(D n, (a,b)) is Z 3 -magic. Example 3.4. Consider the group dihedral group D 0 = {, a, a, a 3, a 4, b, ab, a b, a 3 b, a 4 b} with the generating set A = {a,b} such that a 5 = b =. The Cayley digraph for the dihedral group D 0 and its Z 3 -magic labeling is shown in Figure Figure 3: Z 3 -magic labeling of the Cayley digraph for the dihedral group D 0 Definition 3.5. For any natural number n, we use Z n to denote the additive cyclic group of integers modulo n. In this section we consider the group (Z k,a) where A is a generating set {a,b,b + k} with the property that gcd(a,b,k) = and either gcd(a b,k) or gcd(a, k) = or gcd(b,k) = or both a and k are even or a is odd and either b or k is odd. Algorithm 3.6. Input: The group Z n with the generating set (a,b,b + k).

9 On Z 3 -magic labeling and Cayley digraphs 365 Step : Construct the Cayley digraph Cay(Z n, (a,b,b + k)), using Definition 3.5. Step : Denote the vertex set of Cay(Z n, (a,b,b+k)) as V = {v,v,v 3,...,v n } Step 3 : Denote the edge set of Cay(Z n, (a,b,b + k)) as E(E a,e b,e c ) = {e,e,e 3,...,e 3n } where E a = Set of all out going arcs from v i generated by a E b = Set of all out going arcs from v i generated by b E c = Set of all out going arcs from v i generated by b + k Step 4 : Define f such that f(v i ) = i +, for 0 i n Step 5 : Define g a such that g a (v i ) = ; 0 i n Step 6 : Define g b such that g b (v i ) = ; 0 i n Step 7 : Define g c such that g c (v i ) = ; 0 i n Output: Z 3 -magic labeling for Cayley digraph Cay(Z n, (a,b,b + k)). Theorem 3.7. The Cayley digraph associated with the group Z n with the generating set (a,b,b + k) admits Z 3 -magic labeling. Proof. From the construction of the cayley digraph for the group Z n, we have Cay(Z n, (a,b,b + k)) has n vertices and 3n arcs. Let us denote the vertex set of Cay(Z n, (a,b,b + k)) as V = {v,v,v 3,...,v n } corresponding to the elements {0,,, 3,...,n } of the group Z n respectively and the edge set of Cay(Z n, (a,b,b + k)) as E(E a,e b,e c ) = {e,e,e 3,...,e 3n }. To prove Cay(Z n, (a,b,b + k)) admits Z 3 -magic labeling we have to show that for any vertex v i, the sum of the labels of the outgoing arcs are constant. Define the edges of Cay(Z n, (a,b,b + k)) as follows: For each i Z n, j A define a directed edge v i to v (i+j) (mod n). Consider an arbitrary vertex v i V of the cayley digraph Cay(Z n, (a,b,b + k)). By construction of the cayley digraph we have each vertex has exactly three out going arcs which are from the set E a,e b,e c respectively where E a = {(v i,v (a+i) (mod n) 0 i n } E b = {(v i,v (b+i) (mod n) 0 i n } E c = {(v i,v (b+k+i) (mod n) 0 i n } Now define maps f,g a,g b,g c as defined in steps 4, 5, 6 and 7 of the above algorithm. Thus g a (v i )+g b (v i )+g c (v i ) = ++ = mod 3. Hence the cayley digraph associated with dihedral group Z n admits Z 3 -magic labeling. Example 3.8. Consider the group (Z 8 ; 5; 3; 7) consisting of 8 elements where Z 8 = {0,,, 3, 4, 5, 6, 7} and the generating set A = {5, 3, 7}. The cayley digraph for the group (Z 8 ; 5, 3, 7) and its Z 3 -magic labeling is shown in Figure 4. Example 3.9. Consider the group S 3, consisting of 6 elements. The Cayley digraph for the symmetric group S 3 is shown in Figure 5.

10 366 J. Baskar Babujee and L. Shobana Figure 4: Z 3 -magic labeling of the cayley digraph for the group (Z 8 ; 5, 3, 7) Algorithm 3.0. Input: The symmetric group S k with the generating set (τ,σ). Step : Using Definition 3., construct the Cayley digraph Cay(S k, (τ,σ)). Step : Denote the vertex set of Cay(S k, (τ,σ)) as V = {v,v,v 3,...,v n } where n = k! Step 3 : Denote the edge set of Cay(S k, (τ,σ)) as E(E τ,e σ ) = {e,e,e 3,...,e n } where E τ = Set of all out going arcs from v i generated by τ E σ = Set of all out going arcs from v i generated by σ Step 4 : Define f such that f(v i ) = i, for i n Step 5 : Define g τ such that g τ (v i ) =, for i n Step 6 : Define g σ such that g σ (v i ) =, for i n Output: Z 3 -magic labeling for cayley digraph Cay(S k, (τ,σ)). Theorem 3.. The Cayley digraph associated with symmetric group S k ad- v v v 5 v 4 v 6 v 3 Figure 5: Z 3 -magic labeling of the cayley digraph for symmetric group S 3

11 On Z 3 -magic labeling and Cayley digraphs 367 mits Z 3 -magic labeling. Proof. From the construction of the Cayley digraph for the symmetric group S k, we have Cay(S k, (τ,σ)) has n = k! vertices and n arcs. Let us denote the vertex set of Cay(S k, (τ,σ)) as V = {v,v,v 3,...,v n }. To prove Cay(S k, (τ,σ)) admits Z 3 - magic labeling we have to show that for any vertex v i, the sum of the label of its outgoing arcs are constant. Consider an arbitrary vertex v i V of the Cayley digraph Cay(S k, (τ,σ)). By construction of the Cayley digraph we have each vertex has exactly two out going arcs out of which one arc is from the set E τ and another is from the set E σ, where E τ and E σ are as defined in the above algorithm. Now define maps f,g τ and g σ as defined in steps 4, 5 and 6 of the above algorithm. Hence g τ (v i ) + g σ (v i ) = + 0 mod 3. Hence the Cayley digraph associated with symmetric group S k admits Z 3 - magic labeling. References [] S.B. Akers and B. Krishnamurthy, A group theoretic model for symmetric interconnection networks, IEEE Trans. Comput., 38, (989), [] F. Annexstein, M. Baumslag and A.L. Rosenberg, Group action graphs and parallel architectures, SIAM J. Comput., 9, (990), [3] J. Baskar Babujee and L. Shobana, On Z 3 -Magic Graphs, Proceedings of the International Conference on Mathematics and Computer Science (ICMCS 008), Loyola College, 5-6 July (008), [4] M. Doob, On construction of magic graphs, J. Combin. Theory. Ser. B, 7 (974), [5] M. Heydemann, Cayley graphs and interconnection networks. In: G. Hahn and G. Sabidussi (Eds.), Graph Symmetry: Algebraic Methods and Applications, (997), [6] S-M. Lee, F. Saba, E. Salehi and H. Sun, On V 4 -magic graphs, Congressus Nuumerantuim, 56 (00), [7] R.M. Low and S-M. Lee, On group magic eulerian graphs, JCMCC, 50 (004), [8] Richard M. Low and Sin-Min Lee, On Graph magic eulerian graphs, 6th Midwest conference on combinatorics, cryptography and computing; Carbondale, Illinois (00).

12 368 J. Baskar Babujee and L. Shobana [9] J. Sedlacek, On magic graphs, Math. Slov, 6 (976), [0] J. Sedlacek, Some properties of magic graphs, In: Graphs, Hypergraph and Bloc Syst., Proc. Symp. Comb. Anal., Zielona Gora, (976), [] K. Thirusangu, J. Baskar Babujee and R. Rajeswari, Super vertex (a,d)- antimagic labeling for Cayley digraphs, International Journal of Mtahematics and Applications, (-) (009), 6. [] W.D. Wallis, Magic Graphs, Birkhauser Boston, 00. Received: April, 00