Vertex-antimagic total labelings of graphs

Transcription

1 Vertex-antimagic total labelings of graphs Martin Bača Department of Applied Mathematics Technical University, 0400 Košice, Slovak Republic François Bertault Department of Computer Science and Software Engineering University of Newcastle, NSW 308, Australia J. A. MacDougall Department of Mathematics University of Newcastle, NSW 308, Australia Mirka Miller, Rinovia Simanjuntak and Slamin Department of Computer Science and Software Engineering University of Newcastle, NSW 308, Australia Abstract In this paper we introduce a new type of graph labeling, the (a, d)- vertex-antimagic total labeling, which is a generalization of several other types of labelings. A connected graph G(V, E) is said to be (a, d)-vertex-antimagic total if there exist positive integers a, d and a bijection λ : V E {1,,..., V + E } such that the induced mapping g λ : V W is also a bijection, where W = {w λ (x) x V } = {a, a + d,..., a + ( V 1)d} is the set of weights of vertices in G. Properties of these graphs are studied. How to construct labelings for certain families of graphs are shown. Several conjectures and open problems are proposed. 1

2 1 Introduction All graphs considered here are finite, simple, and undirected. The graph G has vertex set V = V (G) and edge set E = E(G) and we let V = v and E = e. For a general reference for graph theoretic notions, see [1]. A labeling (or valuation) of a graph is any mapping that sends some set of graph elements to a set of numbers (usually positive or non-negative integers). If the domain is the vertex-set or the edge-set, the labelings are called respectively vertex-labelings or edge-labelings. In this paper we deal with the case where the domain is V E, and these are called total labelings. A general survey of graph labelings is found in [10] Various authors, beginning with Sedlácěk [18] have introduced labelings that generalize the idea of a magic square. Magic labelings are one-to-one maps onto the appropriate set of consecutive integers starting from 1, satisfying some kind of constant-sum property. A vertex-magic labeling is one in which the sum of all labels associated with a vertex is a constant independent of the choice of vertex. Edge-magic labelings are defined similarly. Vertexmagic total labelings were first introduced in [15]. Such a labeling is a one-to-one mapping λ : E V {1,,, v + e} with the property that there is a constant k such that at any vertex x λ(x) + λ(xy) = k where the sum is over all vertices y adjacent to x. For any labeling we call the sum of the appropriate labels at a vertex the weight of the vertex, denoted w λ (x); so for vertex-magic total labelings we require that the weight of all vertices be the same, namely k and this number is called the magic number for the labeling. Edge-magic total labelings have been studied recently in [0] and readers are referred to [0] and [15] for more background on these subjects and a standardization of the terminology. Hartsfield and Ringel [11] introduced the concept of an antimagic graph. In their terminology, an antimagic labeling is an edge-labeling of the graph with the integers 1,,..., e so that the weight at each vertex is different from the weight at any other vertex. It is an easy exercise to write down many antimagic labelings for most graphs, so some restriction on the vertex-sums is usually introduced. Thus Bodendiek and Walther [8] defined the concept of an (a, d)-antimagic labeling as an edge-labeling in which the vertex weights

3 form an arithmetic progression starting from a and having common difference d. In this paper we introduce the notions of the vertex-antimagic total labeling and the (a, d)-vertex-antimagic total labeling. For a vertex-antimagic total labeling we label all vertices and edges with the numbers from 1 to v + e and require that the weights of the vertices be all distinct. For an (a, d)-vertexantimagic total labeling we impose the restriction that the vertex weights form an arithmetic progression. More formally, we have: Definition 1 A bijection λ : V E {1,,..., v + e} is called a vertexantimagic total labeling of G = G(V, E) if the weights of vertices w λ (x), x V are pairwise distinct. Definition A bijection λ : V E {1,,..., v + e} is called an (a,d)- vertex-antimagic total labeling (VATL) of G if the set of vertex weights is W = {w λ (x) x V } = {a, a + d,..., a + (v 1)d} for some integers a and d. Unless some further restriction is imposed, vertex-antimagic total labelings are too plentiful to be of much interest. Consequently, in this paper we investigate the basic properties of (a, d)-vertex-antimagic total labelings. We point out connections with some other previously studied types of graph labelings, and construct (a, d)-vertex-antimagic total labelings for certain families of graphs. The paper concludes with several conjectures as well as some open problems suitable for further investigation. General Properties.1 Basic Counting Set M = e + v and let S v be the sum of the vertex labels and S e the sum of the edge labels. Since the labels are the numbers 1,,, M, we have as the sum of all labels: M S v + S e = i = ( ) M

4 If we let w λ (x i ) = a + id, then summing the weights over all vertices addds each vertex label once and each edge label twice, so we get: S v + S e = v (a + (v 1)d). Combining these two equations gives us S e + ( ) ( ) M+1 = va + v d. The edge labels could conceivably receive the e smallest labels or, at the other extreme, the e largest labels, or anything between. Consequently, we have e M i S e i. 1 A similar result holds for S v. Combining these last two equations results in the inequalities ( ) ( ) ( ) ( ) ( ) M+1 + e+1 va + v d M+1 v+1 which restrict the feasible values for a and d. However, for particular graphs, we can often exploit their structure to get considerably stronger restrictions. We note that if δ is the smallest degree in G, then the minimum possible weight on a vertex is at least (δ + 1), consequently a v+1 (δ + 1)(δ + ). Similarly, if is the largest degree, then the maximum vertex weight is no more than the sum of the + 1 largest labels. Thus a + (v 1)d = M i=m +1 i (M + 1).. New Labelings from Old Given one VATL on a graph, it may be possible to construct other VATL s from it. Let λ : V E {1,,..., e + v} be a one-to-one map. We define the map λ on V E by λ (x) = M + 1 λ(x), x V, λ (xy) = M + 1 λ(xy), xy E. 4

5 Clearly, λ is also a one-to-one map from the set V E to {1,,..., e + v}; we say λ is the dual of λ. Theorem 1 The dual of an (a, d)-vertex-antimagic total labeling for a graph G is an (a, d)-vertex-antimagic total labeling for some a and d if and only if G is regular. Proof. Suppose λ is an (a, d)-vertex-antimagic total-labeling for G and W = {w λ (x) x V } = {a, a+d,, a+(v 1)d} is the set of vertex weights of G. Then, for a vertex x V we have w λ (x) = λ (x) + xy E λ (xy) = M + 1 λ(x) + [M + 1 λ(xy)] xy E = (r x + 1)(M + 1) w λ (x), where r x is the number of edges incident to the given vertex x. Clearly, the set W = {w λ (x) x V } consists of an arithmetic progression with difference d = d if and only if r x is constant for every x, that is, if and only if G is regular. Corollary 1.1 Let G be a regular graph of degree r. G has an (a, d)-vertexantimagic total labeling if and only if G has an (a, d)-vertex-antimagic total labeling where a = (r + 1)(M + 1) a (v 1)d. Proof. Let G be a regular graph of degree r and λ be an (a, d)-vertexantimagic total labeling for G. If λ is the dual labeling of λ then for every vertex x V we have w λ (x) = (r + 1)(e + v + 1) w λ (x), where w λ (x) is the weight of the vertex x under the labeling λ. We see that w λ (x) = a + (v 1)d is the maximum value of weight under the labeling λ if and only if w λ (x) = (r + 1)(e + v + 1) a (v 1)d is the minimum value of weight under the labeling λ. Can one use a VATL on a graph G to derive a VATL for a subgraph of G? This seems to be a difficult question in general. The following theorem provides one case in which it is possible. 5

6 Theorem Let G be a regular graph of degree r having an (a, d)-vertexantimagic total labeling (a > r +1) in which the label 1 is assigned to an edge z. Then the graph G {z} has an (a, d)-vertex-antimagic total labeling with a = a r 1. Proof. Assume that G is an r-regular graph and λ is the (a, d)-vertexantimagic total-labeling of G. Define a new mapping µ by µ(x) = λ(x) 1, x V, µ(xy) = λ(xy) 1, xy E. Clearly, the map µ is a one-to-one and the label 0 is assigned to edge xy by µ. Then we have w µ (x) = µ(x) + µ(xy) xy E = λ(x) 1 + [λ(xy) 1] = λ(x) + xy E xy E = w λ (x) r 1, λ(xy) r 1 (where the above summations are taken over all vertices adjacent to x). Clearly, the minimum value of w µ (x) occurs when w λ (x) = a. If we delete the edge xy from G, we obtain a graph G {xy} and the restriction of the mapping µ to G {xy} is an (a r 1, d)-vertex-antimagic total labeling. 3 Relations with other Labelings Kotzig and Rosa [14] defined a magic labeling of graph G(V, E) as a bijection f from V E to {1,,..., v+e} such that for all edges xy, f(x)+f(y)+f(xy) is constant. They proved that K m,n has a magic labeling for all m and n and that C n has a magic labeling for all n 3. Definition 3 A connected graph G = (V, E) is said to be (a, d)-antimagic if there exist positive integers a, d and a bijection f : E {1,,..., e} such that W = {w(v) v V } = {a, a + d,..., a + (v 1)d} is the set of weights of vertices. The map f is called an (a, d)-antimagic labeling of G. 6

7 Definition 4 If there exists a one-to-one map f : E Z + such that all vertices have the same weight w(v), then the graph G is called magic and the map f is called a magic labeling of G. A characterization of regular magic graphs is given in [9]. Several necessary and sufficient conditions for the existence of magic graphs can be found in [1, 13]. Supermagic labeling was defined by Stewart [19]. Definition 5 If there exists a bijection f : E {1,,..., e} such that all vertices have the same weight w(x), then the graph G is called super-magic and the map f is called a super-magic labeling of G. Stewart [19] showed that the complete graph K n is super-magic when n = or n > 5 and n 0 mod 4. Let f : E(K n ) {1,,..., E(K n ) } be the super-magic labeling of K n. For K n we have E(K n ) = n(n 1) and V (K n ) = n. Thus, the sum of edge labels is equal to (n n+)(n n) and the 8 magic constant k is (n n+)(n 1). If we labels all vertices in G with E(K 4 n ) + 1, E(K n ) +,..., V (K n ) + E(K n ) } then this label and f combine to give an (k + E(K n ) + 1, 1) = ( n3 +n+, 1)-vertex-antimagic total labeling. 4 A similar argument applies for any graph G that has a super-magic labeling, and so more generally, we have Theorem 3 Every super-magic graph G is (a, 1)-vertex-antimagic total. From [19] we know that K n and K n,n both have super-magic labelings, and consequently, we have the following two corollaries. Corollary 3.1 If n = or n > 5 and n 0 mod 4 then the complete graph K n has an (a, 1)-vertex-antimagic total labeling. Corollary 3. There is an (a, 1)-vertex-antimagic total labeling for K n,n for all n 3. We know (see [1]) that if n = 4k or n = 4k +, k 1, then there exist two classes of super-magic quartic graphs R n. The next corollary follows from Theorem 3. 7

8 Corollary 3.3 If n = 4k or n = 4k +, k 1, then the quartic graphs R n have an (a, 1)-vertex-antimagic total labeling. Theorem 4 (i) If d > 1 then every (a, d)-antimagic graph G = (V, E) is (a + v + e, d 1)-vertex-antimagic total. (ii) Every (a, d)-antimagic graph G = (V, E) is (a + e + 1, d + 1)-vertexantimagic total. Proof. We assume that graph G is (a, d)-antimagic and d > 1. Let f : E {1,,..., e} be an (a, d)-antimagic labeling of G and W = {w(x) x V } = {a, a + d,..., a + (v 1)d} be the set of weights of vertices of G. Let x i, i = 0,..., (v 1) be the vertex with weight w(x i ) = a + id. Define f, f : V {e + 1, e +,..., e + v} as f (x i ) = e + i + 1 and f (x i ) = v + e + 1 f (x i ). Thus we have the labelings f and f combine to give an (a + e + 1, d + 1)-vertex-antimagic total labeling and f and f to an (a + v + e, d 1)-vertex-antimagic total labeling. Theorem 5 Let G be a graph with total labeling whose vertex labels constitute an arithmetic progression with difference d. Then G has a vertex-magic total labeling with magic constant k if and only if G has a (k + (1 v)d, d)- vertex-antimagic total labeling. Proof. Let λ be a vertex-magic total labeling of G and k is the magic constant for λ. Suppose that, under the labeling λ, the vertex label of G constitute an arithmetic progression with difference d; in other words, {λ(x i ) x i V } = {p + (i 1) d i = 1,,..., v} = {p, p + d,..., p + (v 1) d}, p Z +. Then, under the edge labeling of λ, the weights of vertices constitute an arithmetic progression. Let {w λ (x i ) x i V } = {w λ (x i ) λ(x i ) x i V } Define a new mapping µ by = {k p (i 1) d i = 1,,..., v} = {k p, k p d,..., k p (v 1)d}. 8

9 µ(z) = λ(z) for z E,and µ(x i ) = p + (v i)d for x i V. It can be seen that the weights of vertices, under the new mapping µ, constitute the set W = {w µ (x i ) x i V } = {k + (v + 1 i) d i = 1,,..., v} = {k + (v 1) d, k + (v 3) d,..., k + (1 v) d }, i.e., the weights of vertices constitute an arithmetic progression with difference d and the minimum value of weight is k + (1 v)d. The proof of the converse is similar and is omitted. 4 Paths and Cycles Lemma 1 Every odd cycle, C n, n 3, has a ( 3n+5, )-vertex-antimagic total-labeling. Proof. Wallis et all [0] proved that every odd cycle has an edge-magic totallabeling with magic constant k = 5n+3. For cycles (and only for cycles), an edge-magic total-labeling is equivalent to a vertex-magic total-labeling (see [15]) and, moreover, the vertex labels of the considered vertex-magic totallabeling constitute an arithmetic progression with difference d = 1. Thus, by Theorem 5, the odd cycle C n has a ( 3n+5, )-vertex-antimagic total-labeling. Corollary 1.1 Every odd cycle, C n, n 3, has a ( 5n+5, )-vertex-antimagic total-labeling. Proof. The cycle C n is a -regular graph. By using Lemma 1 and Corollary 1.1, it is simple to verify, that the minimal value of the vertex weight of C n is 5n+5. Now we prove that every cycle C n has an (a, 1)-vertex-antimagic total-labeling for all n 3. 9

10 Theorem 6 For every cycle C n, n 3 there is a (3n+, 1)-vertex-antimagic total-labeling. Proof. Let the cycle C n to be (x 1,..., x n ). If we label the vertices and edges in C n by λ(x i ) = i i = 1,..., n λ(x i x i+1 ) = n i i = 1,..., n 1 λ(x n x 1 ) = n then the induced mapping g λ will be { 4n + 1 i i = 1,..., n 1 g λ (x i ) = 4n + 1 i = n which is, certainly, a one-to-one mapping from V to the set {3n +, 3n + 3,..., 4n + 1}. The path P n can be viewed as a cycle C n with an edge deleted. Lemma If n is odd and n 3, then the path P n has a ( 3n 1, )-vertexantimagic total-labeling. Proof. The odd cycle C n from Lemma 1 is a -regular graph having a ( 3n+5, )-vertex-antimagic total-labeling in which the label 1 is assigned to an edge z E. By Theorem we have that the path P n has a ( 3n 1, )- vertex-antimagic total-labeling. Theorem 7 The path P n has a (n+1, 1)-vertex-antimagic total-labeling for any n. Proof. Consider the vertices in P n to be x 1,..., x n and the set of edges is E(P n ) = {x i x i+1 i = 1,..., n 1}. Then label all vertices and edges by n + 1 i = n + i i = 3,..., n 1 λ(x i ) = n i = n n + 1 i = 1 1 i = n 1 λ(x i x i+1 ) = i i =,..., n n i = 1 10

11 By that labeling we have the induced mapping g λ 3n + i i = n 1, n g λ (x i ) = n i i =,..., n 3n + 1 i = 1 Hence, the g λ is a one-to-one mapping from V to the set {n + 1, n +,..., 3n} and certainly, f is a (n + 1, 1)-vertex-antimagic total-labeling. 5 References References [1] M. Bača, I. Holländer, and Ko-Wei Lih, Two classes of super-magic quartic graphs, JCMCC 13 (1997), [] M. Bača and I. Holländer, On (a, d)-antimagic prisms, Ars. Comb. 48 (1998), [3] M. Bača, Antimagic labelings of antiprisms, JCMCC, to appear. [4] M. Bača, Special face numbering of plane quartic, Ars. Comb., to appear. [5] M. Bača and M. Miller, Antimagic face labeling of convex polytopes based on biprisms, JCMCC, submitted. [6] M. Bača, Face antimagic labelings of convex polytopes, Utilitas Math., to appear. [7] M. Bača, Consecutive magic labeling of generalized Petersen graphs, Utilitas Math., submitted. [8] R. Bodendiek and G. Walther, Arithmetisch antimagische graphen, in K. Wagner and R. Bodendiek, Graphentheorie III, BI-Wiss. Verl., Mannheim, [9] M. Doob, Generalizations of magic graphs, J. Comb. Theory B 17, (1974), [10] J.A. Gallian, A Dynamic survey of graph labeling, Electronic. J. Comb 5 (1998), #DS6. 11

12 [11] N. Hartsfield and G. Ringel, Pearls in Graph Theory, Academic Press, Boston-San Diego-New York-London, [1] R.H. Jeurissen, Magic graphs, a characterization, Report 801, Mathematisch Instituut, Katholieke Universiteit Nijmegen, 198. [13] S. Jezný and M. Trenklér, Charaterization of magic graphs, Czechoslovak Math. J. 33 (1983), [14] A. Kotzig and A. Rosa, Magic valuations of finite graphs, Canad. Math. Bull. 13 (1970), [15] J.A. MacDougall, M. Miller, Slamin, and W.D. Wallis, Vertex-magic total labelings of graphs, submitted. [16] M. Miller and M. Bača, Antimagic valuations of generalized Petersen graphs, Australasian J. Comb., submitted. [17] M. Miller, M. Bača, and Y. Lin, On two conjectures concerning (a, d)- antimagic labelings of antiprisms, JCMCC, submitted. [18] J. Sedlácěk, Problem 7 in Theory of Graphs and its applications, Proc. Symp. Smolenice, June 1963, Praha (1964), p. 16. [19] B.M. Stewart, Supermagic complete graphs, Can. J. Math. 19 (1967), [0] W.D. Wallis, E.T. Baskoro, M. Miller, and Slamin, Edge-magic total labelings of graphs, submitted. [1] D.B. West, An Introduction to Graph Theory, Prentice-Hall,