Note on group distance magic complete bipartite graphs

Transcription

1 Cent. Eur. J. Math. 12(3) DOI: /s z Central European Journal of Mathematics Note on group distance magic complete bipartite graphs Research Article Sylwia Cichacz 1 1 Faculty of Applied Mathematics, AGH University of Science and Technology, Al. Miciewicza 30, Kraów, Poland Received 3 April 2013; accepted 9 July 2013 Abstract: A Γ-distance magic labeling of a graph G = (V, E) with V = n is a bijection l from V to an Abelian group Γ of order n such that the weight w(x) = y N G (x) l(y) of every vertex x V is equal to the same element µ Γ, called the magic constant. A graph G is called a group distance magic graph if there exists a Γ-distance magic labeling for every Abelian group Γ of order V (G). In this paper we give necessary and sufficient conditions for complete -partite graphs of odd order p to be Z p -distance magic. Moreover we show that if p 2 (mod 4) and is even, then there does not exist a group Γ of order p such that there exists a Γ-distance labeling for a -partite complete graph of order p. We also prove that K m,n is a group distance magic graph if and only if n + m 2 (mod 4). MSC: 05C25, 05C78 Keywords: Graph labeling Abelian group Versita Sp. z o.o. 1. Introduction All graphs considered in this paper are simple finite graphs. We use V (G) for the vertex set and E(G) for the edge set of a graph G. The neighborhood N(x) of a vertex x is the set of vertices adjacent to x, and the degree d(x) of x is N(x), the size of the neighborhood of x. A distance magic labeling (also called sigma labeling) of a graph G = (V, E) of order n is a bijection l : V {1, 2,..., n} with the property that there is a positive integer µ (called the magic constant) such that y N G (x) l(y) = µ for every x V. If a graph G admits a distance magic labeling, then we say that G is a distance magic graph [1]. The sum l(y) is called the weight of the vertex x and denoted by w(x). y N G (x) cichacz@agh.edu.pl 529

2 Note on group distance magic complete bipartite graphs The concept of distance magic labeling has been motivated by the construction of magic squares. Observe also that constant sum partition of {1, 2,..., n} leads to complete multipartite distance magic labeled graphs. For instance, the partition {1, 4}, {2, 3} of the set {1, 2, 3, 4} with constant sum 5 leads to distance magic labeling of the complete bipartite graph K 2,2, see [2]. Beena proved Theorem 1.1 ([2]). Let m and n be two positive integers such that m n. The complete bipartite graph K m,n is a distance magic graph if and only if m + n 0 or 3 (mod 4) and either n (1 + 2)m 1/2 or 2(2n + 1) 2 (2m + 2n 1) 2 = 1. Fronče in [5] defined the notion of group distance magic graphs, i.e. the graphs allowing the bijective labeling of vertices with elements of an Abelian group resulting in constant sums of neighbor labels. Definition 1.2. A Γ-distance magic labeling of a graph G = (V, E) with V = n is a bijection l from V to an Abelian group Γ of order n such that the weight w(x) = y N G (x) l(y) of every vertex x V is equal to the same element µ Γ, called the magic constant. A graph G is called a group distance magic graph if there exists a Γ-distance magic labeling for every Abelian group Γ of order V (G). The connection between distance magic graphs and Γ-distance magic graphs is as follows. Let G be a distance magic graph of order n with the magic constant µ. If we replace the label n in a distance magic labeling for the graph G by the label 0, then we obtain a Z n -distance magic labeling for the graph G with the magic constant µ µ (mod n). Hence every distance magic graph with n vertices admits a Z n -distance magic labeling. Although a Z n -distance magic graph on n vertices is not necessarily a distance magic graph (see [3, 5]). Notice that constant sum partition of a group Γ leads to complete multipartite Γ-distance magic labeled graphs. For instance, the partition {0}, {1, 2, 4}, {3, 5, 6} of the group Z 7 with constant sum 0 leads to a Z 7 -distance magic labeling of the complete tripartite graph K 1,3,3. Kaplan at al. proved Theorem 1.3 ([6]). Let n = r 1 + r r q be a partition of the positive integer n, where r i 2 for i = 1, 2,..., q. Let A = {1, 2,..., n}. Then the set A can be partitioned into pairwise disjoint subsets A 1, A 2,..., A q such that for every 1 i q, A i = r i with a A i a 0 (mod n + 1) if n is even and a A i a 0 (mod n) if n is odd. We use Theorem 1.3 to give necessary and sufficient conditions for complete -partite graphs of odd order p to be Z p -distance magic. Moreover we prove that K m,n is a group distance magic graph if and only if n + m 2 (mod 4). This wor will also be potentially useful for group theorists woring on Abelian groups. 2. Preliminaries Assume Γ is an Abelian group of order n with the operation denoted by + and the identity element a 0. For convenience we will write a to denote a + a + + a (where the element a appears times), a to denote the inverse of a and we will use a b instead of a + ( b). The fundamental theorem of finite Abelian groups states that a finite Abelian group Γ of order n can be expressed as the direct sum of cyclic subgroups of prime-power order. This implies that Γ = Z p α 1 1 Z p α 2 2 Z α p, where n = p α 1 1 pα pα 530

3 S. Cichacz and p i for i {1, 2,..., } are not necessarily distinct primes. In particular, if n 2 (mod 4), then Γ = Z 2 A for some Abelian group A of odd order n/2. Recall that any group element ι a 0 such that 2ι = a 0 is called an involution, and that a non-trivial finite group has involutions if and only if the order of the group is even. Moreover, every cyclic group of even order has exactly one involution. If t is the number of these cyclic components whose order is a power of 2, i.e. Γ = Z 2 α 1 Z 2 α 2 Z 2 αt Z p β 1 1 and primes p i > 2 for i = 1, 2,...,, then Γ has 2 t 1 involutions. Let s(γ) = g Γ g. The following lemma was proved in [4]. Z β p 2 Z β 2 p Lemma 2.1 ([4, Lemma 8]). Let Γ be an Abelian group. (i) If Γ has exactly one involution ι, then s(γ) = ι. (ii) If Γ has no involutions, or more than one involution, then s(γ) = a Group distance magic graphs K m,n We start with the following observation: Observation 3.1. Let G = K n0,n 1,n 2,...,n 1 be a complete -partite graph such that 1 n 0 n 1... n 1 and n = n 0 + n n 1 is odd. The graph G is a Z n -distance magic graph if and only if n i 2 for i = 1, 2,..., 1. Proof. Let G have the partition vertex sets V i such that V i = n i for i = 0, 1,..., 1. Assume first that n 0 = n 1 = 1. Let x V 0 and y V 1. Suppose that the graph G is Z n -distance magic and that l is a Z n -distance magic labeling of G, then w(x) = g Z n g l(x) = w(y) = g Z n g l(y). Thus l(y) = l(x), a contradiction. If n i 2 for i = 0, 1,..., 1 the set A = {1, 2,..., n} can be partitioned into pairwise disjoint A 0,..., A 1 such that for every 0 i 1, A i = n i with a A i a 0 (mod n) by Theorem 1.3. Label the vertices from a vertex set V i using elements from the set A i for i = 0, 1,..., 1. If now n 0 = 1, then the set A = {1, 2,..., n 1} can be partitioned into pairwise disjoint subsets A 1, A 2,..., A 1 such that for every 1 i 1, A i = n i with a A i a 0 (mod n) by Theorem 1.3. Let A 0 = {0}. As above, label the vertices from a vertex set V i using elements from the set A i for i = 0, 1,..., 1. We obtain that w(x) = 0 (mod n) for all x V (G). Theorem 3.2. Let G = K n0,n 1,n 2,...,n 1 be a complete -partite graph and n = n 0 + n n 1. If n 2 (mod 4) and is even, then there does not exist an Abelian group Γ of order n such that G is a Γ-distance magic graph. Proof. Assumption n 2 (mod 4) implies that Γ = Z 2 A for some Abelian group A of odd order n/2 and there exists exactly one involution ι Γ. Let G have the partition vertex sets V i such that V i = n i for i = 0, 1,..., 1. Suppose that l is a Γ-distance magic labeling of the graph G. It is easy to observe that v V i l(v) = γ for i = 0, 1,..., 1, for some element γ Γ. Thus γ = 1 i=0 v V i l(v) = g Γ g = s(γ) = ι by Lemma 2.1. Since n/2 is odd and Γ = Z 2 A, such an element γ Γ does not exist, a contradiction. 531

4 Note on group distance magic complete bipartite graphs Notice that the assumption that is even in Theorem 3.2 is necessary. For instance, the partition {3}, {1, 2}, {0, 4, 5} of the group Z 6 with constant sum 3 gives a Z 6 -distance magic labeling of the complete tripartite graph K 1,2,3 with the magic constant µ = 0. Theorem 3.3. The complete bipartite graph K m,n is a group distance magic graph if and only if m + n 2 (mod 4). Proof. We can assume that m + n 2 (mod 4) by Theorem 3.2. Let K m,n have the partition vertex sets A = {x 0, x 1,..., x m 1 } and B = {y 0, y 1,..., y n 1 }. Let Γ = {a 0, a 1,..., a m+n 1 }. Recall that by a 0 we denote the identity element of Γ. Notice that if l is a Γ-distance magic labeling for K m,n, then the magic constant µ = m 1 i=0 l(x i) = n 1 j=0 l(y j). Suppose first that m + n is odd, then without loss of generality we can assume that m is odd and n is even. Moreover, since there is no involution in Γ we can assume that a i+1 = a i for i = 1, 3, 5,..., m + n 2. Label the vertices of K m,n in the following way: l(x i ) = a i, i = 0, 1,..., m 1, l(y j ) = a m+j, j = 0, 1,..., n 1. j=0 l(y j) = a 0. From now on we will assume that m + n 0 (mod 4). We will consider two cases on the number of involutions in Γ. Case 1: There exists exactly one involution ι Γ. Since n + m 0 (mod 4) the group Γ = Z α 2 A for some α > 1 and some Abelian group A of odd order n/2 α (with the identity element a 0 A). Thus there exists a subgroup {a 0, a, 2a, 3a} of Γ, where a = (2 α 2, a 0 ) and 2a = (2α 1, a 0 ) is the involution ι Γ. Without loss of generality we can assume that a 1 = a, a 2 = 2a, a 3 = 3a and a i+1 = a i for i = 4, 6, 8,..., m + n 2. Case 1.1: m 1 (mod 4) and n 3 (mod 4). l(x 0 ) = a 3, l(x i ) = a i+3, i = 1, 2,..., m 1, l(y 0 ) = a 0, l(y 1 ) = a 1, l(y 2 ) = a 2, l(y j ) = a m+j, j = 3, 4..., n 1. j=0 l(y j) = a 3. Case 1.2: m, n 2 (mod 4). j=0 l(y j) = a 3. l(x 0 ) = a 0, l(x 1 ) = a 3, l(x i ) = a i+2, i = 2, 3,..., m 1. l(y 0 ) = a 1, l(y 1 ) = a 2, l(y j ) = a n+j, j = 2, 3,..., n 1. Case 2: There exist t > 1 involutions ι 1, ι 2,..., ι t Γ. Recall that t is odd. Without loss of generality we can assume that a j = ι j for j = 1, 2,..., t and a j+1 = a j for j = t + 1, t + 2,..., m + n 2. Case 2.1: m, n 2 (mod 4). l(x i ) = a i, i = 0, 1,..., m 1, l(y j ) = a m+j, j = 0, 1,..., n 1. j=0 l(y j) = a 0 if m t + 1 by Lemma 2.1. If m < t + 1, then recall that {a 0, ι 1, ι 2,..., ι t } is a subgroup of Γ and thus m 1 i=0 l(x i) = a j0 for some 0 j 0 t. Furthermore n 1 j=0 l(y j) = s(γ) m 1 i=0 l(x i) = a 0 a j0 = a j0 = a j0, by Lemma 2.1. Case 2.2: m 1 (mod 4) and n 3 (mod 4). l(x i ) = a i+1, i = 0, 1,..., m 1, l(y 0 ) = a 0, l(y j ) = a m+j, j = 1, 2,..., n 1. As above, m 1 j=0 l(y j) = a j0 for some 0 j 0 t. 532

5 S. Cichacz Acnowledgements The author was supported by National Science Centre grant nr 2011/01/D/ST/ References [1] Arumugam S., Fronce D., Kamatchi N., Distance magic graphs a survey, J. Indones. Math. Soc., 2011, Special edition, [2] Beena S., On Σ and Σ labelled graphs, Discrete Math., 2009, 309(6), [3] Cichacz S., Note on group distance magic graphs G[C 4 ], Graphs Combin. (in press), DOI: /s z [4] Combe D., Nelson A.M., Palmer W.D., Magic labellings of graphs over finite abelian groups, Australas. J. Combin., 2004, 29, [5] Fronce D., Group distance magic labeling of Cartesian product of cycles, Australas. J. Combin., 2013, 55, [6] Kaplan G., Lev A., Roditty Y., On zero-sum partitions and anti-magic trees, Discrete Math., 2009, 309(8),