Group Distance Magic Labeling for the Cartesian Product of Cycles

Transcription

1 Group Distance Magic Labeling for the Cartesian Product of Cycles by Stephen Balamut MS Candidate: Applied and Computational Mathematics Advisor: Dalibor Froncek Department of Mathematics and Statistics University of Minnesota Duluth

2 Abstract Over the past half century, graph labeling has been an emerging area of graph theory. There are still many open problems. To this end, I have worked toward some new results related to distance magic labeling. In this paper, I first outline a number of graph labelings with some interesting results. Secondly, I will introduce two new results on distance magic labeling in a group. Of the two results, I demonstrate a distance magic labeling for a special class of graphs when the sums are calculated using modular arithmetic, and I show that a related case is not group distance magic. i

3 Contents 1 INTRODUCTION Magic Square Basic Definitions Graph Products Graph Labelings RELATED RESULTS Magic Labeling Edge-Magic Total Labeling Super Edge-Magic Labeling Vertex-Magic Total Labeling Super Vertex-Magic Total Labeling Totally Magic Labeling Anti-Magic Labeling Vertex-Antimagic Total Labeling Distance Magic Labeling MAIN RESULTS 11 4 CONCLUSION 18 ii

4 List of Figures 1 Magic Square (n = 3) Magic Square (n = 4) Examples of the Three Fundamental Graph Products [18] Small Cases of Distance Magic [23] C 4 C A C 4 C 4 grid representation A different view of C 4 C C 8 C 4 with Indexing C 4 C 4 Demonstrating Neighborhood Sums C 8 C 4 Example of Labeling Main Superdiagonal C 8 C 4 Example of Labeling C 7 C 5 Demonstrating Layout iii

5 1 INTRODUCTION 1.1 Magic Square Much of what is described in this paper is linked to the 3,000 year old concept of the magic square. This is a square n by n grid of the integers 1 to n 2 where each row, each column, and both diagonals sum to the same constant. The ideas here have been linked to the concepts in a number of labelings, and, along with the spirit of the puzzle, the ideas used here inspire expansions of the problem. For example, in order to find the sum of a row or column, one can use the summation formula for the first n 2 numbers and divide by n. That is, n2 (n 2 1) 2n = n(n2 1) 2 is the magic constant. Several methods have been devised to number magic squares of increasingly large size including using smaller magic squares to make larger ones. Below are some small examples [26] Figure 1: Magic Square (n = 3) Figure 2: Magic Square (n = 4) Transitioning, let us look at this notion from a graph theory point of view. The following section includes some background on graph theory and descriptions of several related concepts that will eventually lead to some correlated results. 1.2 Basic Definitions I will be following the notation and terminology in the book A First Look at Graph Theory by Clark and Holton [5]. Any topics not covered in the book will have notation based on the specific paper cited or a previously cited paper. Definition 1.1 (Clark, Holton [5]). A graph G=(E(G), V (G)) consists of two finite sets. The vertex set of the graph V (G) is a non-empty set of elements called vertices. The edge set of the graph E(G) is a set of the elements called edges and is possibly empty. Each edge e in E is assigned an unordered pair of vertices (u, v), called the end vertices of e [5]. Definition 1.2 (Clark, Holton [5]). The vertices (sometimes called nodes) are said to be joined by edges. A vertex joined to an edge is said to be incident to the edge. The way that edges connect vertices in a graph is unrestricted. Definition 1.3 (Clark, Holton [5]). When an edge starts and ends at the same vertex, the edge is called a loop. If two edges have the same starting and ending points, the two edges are called parallel. Graphs with neither loops nor parallel edges are called simple graphs. 1

6 In graph theory, simple graphs are studied most often. In this project we will only deal with simple graphs. From now on, unless otherwise specified, it is assumed that a graph is simple [5]. Definition 1.4 (Clark, Holton [5]). An edge e of a graph G is said to be incident with the vertex v if v is an end vertex of e. Two edges e and f incident with a common vertex are said to be adjacent. Similarly, two vertices u and v incident with the same edge are said to be adjacent or neighbors. Definition 1.5 (Clark, Holton [5]). The set of vertices that are adjacent to a vertex v is known as the neighborhood set of vertex v and is denoted N(v). Definition 1.6 (Clark, Holton [5]). The degree d(v) of a vertex v in a graph G is the number of edges incident with v. That is, it is the number of times v is an end vertex of an edge. A vertex is either odd or even based on the degree of the vertex. Definition 1.7 (Clark, Holton [5]). For some positive integer k, a graph G with d(v) = k for each v in G is called a k-regular graph. A regular graph is k-regular for some k. Definition 1.8 (Clark, Holton [5]). A walk is a sequence of alternating vertices and edges that are incident with the element directly before and after. A walk begins and ends at vertices and passes through a sequence of adjacent vertices on the way. If a walk starts and ends at the same vertex, it is called closed; otherwise it is open. If the edges of a walk are distinct, then it is called a trail. If the vertices of a walk are distinct, then it is called a path and is denoted P n where n is the number of vertices in the path. Definition 1.9 (Clark, Holton [5]). A vertex v is said to be connected to a vertex u in a graph G if there is a path from v to u. A graph is said to be connected if each pair of vertices in G are connected. A graph that is not connected is called disconnected. Definition 1.10 (Clark, Holton [5]). A complete graph K n is a graph in which each vertex is adjacent to every other vertex in the graph. Definition 1.11 (Clark, Holton [5]). A nontrivial closed trail in a graph G is called a cycle C n if its origin and internal vertices are all distinct where n is a positive integer denoting the number of vertices. Definition 1.12 (Clark, Holton [6]). A tree is a connected graph that contains no cycles. A cycle is a 2-regular graph. Later, we will define a graph using the product of cycles that will be 4-regular. Mentioned first, though, will be a graph called a hypercube (or k-cube). Definition 1.13 (Clark, Holton [5]). A hypercube (or k-cube) is a graph whose vertices are ordered k-tuples of 1 s and 0 s such that two vertices are adjacent when the two k-tuples differ in precisely one position. Some of the related results presented below will reference a type of graph called bipartite. This type of graph can be used to match elements of one group to elements of another. Definition 1.14 (Clark, Holton [5]). Let G be a graph. Then G is bipartite if the vertex set V of G can be partitioned into two non-empty subsets X and Y in such a way that each edge has one end in X and one end in Y. A complete bipartite graph K m,n where m and n are the numbers of vertices in X and Y is a bipartite graph where every vertex in X is adjacent to every vertex in Y. 2

7 1.3 Graph Products There are three fundamental graph products: Cartesian, direct, and strong. The graphs G and H are said to be the factors of the product graph. The Cartesian product will be the focus of this paper and will be represented with the symbol. The two additional products of graphs are also studied in graph theory frequently and are also defined below [18]. Definition 1.15 (Hammack, Imrich, Klavzar [18]). Define the Cartesian product G H to be a graph with vertex set V (G) V (H) and (g,h) is adjacent to (g,h ) if and only if (i) g=g and hh E(H) or (ii) h=h and gg E(G). Definition 1.16 (Hammack, Imrich, Klavzar [18]). The direct product of graphs can be defined as G H and with the vertex set V (G) V (H) such that the vertices (g,h) and (g,h ) are adjacent exactly when gg E(G) and hh E(H). Definition 1.17 (Hammack, Imrich, Klavzar [18]). The strong product of graphs can be defined as G H such that V (G H)={(g, h) g V (G) and h V (H)} and E(G H)=E(G H) E(G H). Figure 3 shows an example of each product. The graphs of the factors are included to show how each is constructed. Figure 3: Examples of the Three Fundamental Graph Products [18] 1.4 Graph Labelings Graph labelings are ways to assign integers to vertices and/or edges of a graph under some rule or restriction. This area of mathematics has only been around for about a half century and has already been published in more than 1,000 papers. A collection of these results has been gathered by Joseph Gallian in A Dynamic Survey of Graph Labeling [12]. Definition A graph labeling is an assignment f of integers to vertices and/or edges of a graph. Research on labelings tends to focus on classes of graphs that can be generalized in some way. Simple graphs (graphs with no parallel edges or loops), regular graphs (graphs with the same degree at each vertex), symmetric graphs, and vertex-transitive graphs (graphs where the vertices are indistinguishable from those around it) are all examples of this. For the results in this paper, graphs that are products of cycles will be the focus. Before looking at a number of magic type labelings, I will first introduce some of the Rosa types of graph labelings, which were among the first to be defined. 3

8 Definition Let G be a graph with n edges. Then a ρ-labeling of G is a one-to-one function f: V (G) {0, 1,..., 2n} such that {min{ f(u) f(v), 2n+1 f(u) f(v) } : {u, v} E(G)}= {1, 2,..., n} [8]. As we will see with graceful labelings, the difference of vertex labels is used to determine the edge label, but this time the labels could be the difference subtracted from 2n 1. This next labeling and its more restrictive variations described below were introduced by Rosa in This labeling was originally called β-valuation by Rosa. Now it is referred to as graceful labeling. Assigning integers to each vertex and each edge, this type of labeling has assigned values for edges based on the value of their incident vertices. Definition A graceful labeling is a ρ-labeling except the vertex labels are in {0, 1, 2,..., n} and the value of each edge, uv, is f(u) f(v) [13]. In other words, in order for a graph to be considered graceful, there still needs to be an injection from the first n integers to the edges where n is the number of edges. Most labelings have their origins in this type [13]. An interesting special case of the graceful labeling is the α-labeling. Definition An α-labeling is a graceful labeling with the additional condition that there exists some constant k such that the following is true for each edge xy: f(x) k < f(y) or f(y) k < f(x). This is a way to balance a graph labeling and is unsurprisingly sometimes called balanced or stronglygraceful [14]. Finally, a harmonious labeling is an additive analogy of a graceful labeling. It is identical to graceful labeling except edge labels are computed f(x) + f(y) (mod n) where n is the number of edges. Thus, to calculate the label of a particular edge, simply compute the sum of the incident vertices modulo n [12]. Next, I will describe some related work done with graph labeling. At the end of the section I will introduce the topic and paper that inspired my new results. 2 RELATED RESULTS There are numerous types of magic labelings in graph theory. Each has its own conditions based on the labels at each vertex and/or edge. The following are some different types of magic labelings that have been studied along with some interesting results for each. 2.1 Magic Labeling In the 1960 s magic labelings began being studied. Naturally, a number of variations were created. A graph is said to be semi-magic if the edges can be labeled in a way that the sum of the incident edges is the same for every vertex chosen. A semi-magic graph becomes a magic graph when the edges are labeled with distinct positive integers. As might be expected, a supermagic graph is a magic graph in which the edges are consecutive positive integers [15]. Ivanco showed the following. Theorem 2.1 (Ivanco [20]). C 2n C 2k, n, k 2 is supermagic. Theorem 2.2 (Ivanco [20]). C n C n for each integer n 3 is supermagic. 4

9 Theorem 2.3 (Ivanco [20]). Q n is supermagic if either n=1 or n 4 and n 0 (mod 2). He also conjectured the following result that deals with the Cartesian products of cycles. Conjecture 2.4 (Ivanco [20]). C n C k is supermagic. 2.2 Edge-Magic Total Labeling An edge-magic labeling is called edge-magic total (abbreviated EMT) to distinguish from other magic type labelings. It is a bijection f from the set V E to the set of integers, {1,2,, V + E } such that for every edge uv, f(u) + f(v) + f(uv) = h where h is a constant. We mentioned that for a 2-regular graph, an EMT labeling is the same as a vertex magic total labeling as described below [15]. When studying graph labelings, mathematicians often work with certain types or classes of graphs. In the spirit of this, Ngurah, Baskoro, Simanjuntak, and Uttunggadewa studied kc 4 -snakes. Definition 2.1 (Ngurah, Baskoro, Simanjuntak, Uttunggadewa [24]). A kc 4 -snake is a connected graph with k blocks where the blocks are repeated components of the graph. Each of the blocks is isomorphic to the cycle C 4 such that the block-cut-vertex graph is a path (a graph whose vertices correspond to the the blocks and are adjacent if the blocks were adjacent). One could think of this as a row of diamonds. Theorem 2.5 (Ngurah, Baskoro, Simanjuntak, Uttunggadewa [24]). kc 4 -snakes are EMT. Kotzig and Rosa also showed some interesting results. Theorem 2.6 (Kotzig, Rosa [15]). Complete bipartite graphs K m,n are EMT for all m, n. Theorem 2.7 (Kotzig, Rosa [15]). Cycles C n are EMT for n 3. Wallis, Baskoro, Miller, and Slamin showed some other interesting results. Theorem 2.8 (Wallis, Baskoro, Miller, Slamin [27]). Odd cycles C n have EMT labelings for the following magic constants: (5n + 3) (7n + 3) 3. 3n n + 2 Theorem 2.9 (Wallis, Baskoro, Miller, Slamin [27]). Even cycles C n have EMT labelings for the magic constant 1 2 (5n + 4). Theorem 2.10 (Wallis, Baskoro, Miller, Slamin [27]). Cycles C n of length divisible by 4 have EMT labelings for the following magic constants: (7n + 2) 2. 3n 3. 3n + 3 They also proved a result about kites. 5

10 Definition 2.2 (Wallis, Baskoro, Miller, Slamin [27]). A kite is a cycle with a path connected to one of its vertices. A kite of tail length 1 is a cycle with a path of length 1 connected to one of the vertices in the cycle. Theorem 2.11 (Wallis, Baskoro, Miller, Slamin [27]). Kites with tail length 1 are EMT. Perhaps the most interesting conjecture about EMT labelings is the following. Although no one has been able to prove or disprove it, the conjecture continues to be a popular unsolved problem. Conjecture 2.12 (Baca, Miller [2]). Every tree is EMT. 2.3 Super Edge-Magic Labeling A super edge-magic labeling (SEM) is the same as the edge-magic total labeling with the additional condition that the vertices map to the smallest integers {1, 2,..., V }. Below, super vertex-magic total labelings will have the same additional condition. Enomoto, Llado, Nakamigawa, and Ringel showed the following about SEM labeling. Theorem 2.13 (Enomoto, Llado, Nakamigawa, Ringel [9]). Cycles C n are SEM if and only if n is odd. Definition 2.3 (Clark, Holton [5]). A wheel graph W n is a graph made of a cycle C n 1 and a single vertex where there is a path of length 1 from each vertex in the cycle to the single vertex. I will not that other authors will define a wheel graph using the cycle C n rather than C n 1. Theorem 2.14 (Enomoto, Llado, Nakamigawa, Ringel [9]). There does not exist an n such that the wheel graph W n is super edge-magic. 2.4 Vertex-Magic Total Labeling A vertex-magic total labeling (abbreviated VMT) is a bijection from the set V E to {1, 2,..., V + E }. For this labeling there is some constant weight h such that for each vertex x the sum of the label for x and the labels for all the edges incident to x is h [2]. Related to the results later in this paper, Froncek, Kovar, and Kovarova showed the following about the Cartesian product of cycles. Later it will be shown that this result does not necessarily hold for similar magic type labelings; specifically, products of two odd cycles are not distance magic. Theorem 2.15 (Froncek, Kovar, Kovarova [11]). For each m, n 3 and at least one of m, n odd, there exists a VMT labeling of C m C n for each of the following magic constants: (15mn + m + 4) (17mn + 5) One open problem in this area comes from the paper Vertex-Magic Labelings of Regular Graphs by MacDougal [22]. Since the conjecture was made, a number of constructions for regular graphs have been formulated, but it has never been proven. Conversely, no counterexample has been found either. Conjecture 2.16 (Baca, Miller [2]). Every regular graph has a VMT labeling besides K 2 and 2K 3 Petr Kovar has since put forward a number of results working toward this conjecture. Although he does not cover every case, he was able to show that many regular graphs are VMT. The following theorems demonstrate this. 6

11 Theorem 2.17 (Kovar [21]). Let G be a (2 + s)-regular graph such that it contains an s-regular factor G which allows a VMT labeling with magic constant h and vertex labels being consecutive integers starting at k. Then G is VMT. Theorem 2.18 (Kovar [21]). Let G be a (2r +s)-regular graph such that it contains an s-regular factor G which allows a VMT labeling with magic constant h and vertex labels being consecutive integers starting at k. Then G is VMT. 2.5 Super Vertex-Magic Total Labeling Similar to the vertex-magic total labeling, a super vertex-magic total labeling is one with the additional condition that the smallest integers are the vertex labels and the largest are the edge labels. That is, the vertices are labeled using the set {1, 2,..., V }. Some interesting results are given in a paper by MacDougal, Miller, and Sugeng [22]; however, as is expected they are more restricted under the new condition. Theorem 2.19 (MacDougal, Miller, Sugeng [22]). A cycle C n has a super vertex magic total labeling if and only if n is odd. They also showed that a number of different graphs do not have a super VMTL such as complete bipartite graphs and graphs with a vertex of degree one (including trees) [22]. Below, we will see a negative result for distance magic labelings. 2.6 Totally Magic Labeling A totally magic graph is one in which it has a vertex-magic total labeling and an edge-magic total labeling. Remembering that each vertex is dependent on the incident edges and each edge is dependent on its incident labels, this is obviously a difficult labeling. Many results are small special cases so far. Exoo, Ling, McSorley, Phillips, and Wallis studied how cycles and vertices of degree one within a graph can affect the way the graph must be labeled. Theorem 2.20 (Exoo, Ling, McSorley, Phillips, and Wallis [10]). The only totally magic cycle is C 3 (or K 3 ). Theorem 2.21 (Exoo, Ling, McSorley, Phillips, and Wallis [10]). The only connected totally magic graph containing a vertex of degree one is P 3. Consequently, the only totally magic trees are P 3 and K 1. Theorem 2.22 (Exoo, Ling, McSorley, Phillips, and Wallis [10]). If a totally magic graph contains a triangle (C 3 ), then the sum of the edge labels incident to any one vertex of the triangle and outside the triangle is the same no matter which vertex is chosen. Now, to take a look at a type of labeling that assigns labels to the vertices similarly to magic labeling, I have included anti-magic labeling. In this case, the concentration shifts from equality to pairwise distinct. 7

12 2.7 Anti-Magic Labeling An anti-magic labeling of a graph is essentially the opposite of a magic labeling. That is, it is a bijection from the vertices to consecutive integers, but the sums found must be distinct. Each vertex sum must be different from every other vertex sum [28]. Wang and Hsiao showed a number of results including the Cartesian products of basic graphs. As we saw with some other magic type labelings, this type of product will continue to be part of the theme of this paper. Theorem 2.23 (Wang Hsiao [28]). P m P n (lattice grid graphs) are antimagic for m n 2. Expanding, they also showed a result including a cycle as one of the factors. Theorem 2.24 (Wang Hsiao [28]). C m P n (prism grids) are antimagic. Theorem 2.25 (Wang Hsiao [28]). Other antimagic graphs include 1. Paths P n for n 3 2. Cycles C n for n 3 3. Hypercube graphs Q n A strong result in this area was shown by Cheng. Theorem 2.26 (Cheng [4]). All Cartesian products of two or more regular graphs of positive degree are antimagic. This result about antimagic graphs is very interesting since nothing like this has been shown for distance magic graphs. After describing a number of graphs that are antimagic, Hartsfield and Ringel had the following conjecture. Conjecture 2.27 (Hartsfield, Ringel [19]). Every connected graph, excluding K 2, is antimagic. A number of researchers have unsuccessfully tried to show this to be false. At the same time, given a connected graph, it is typically not difficult to find an antimagic labeling for it. Moreover, there are often multiple such labelings. With that in mind, as with magic labelings, more restrictions have been added to form new types of anti-magic graphs [2]. 2.8 Vertex-Antimagic Total Labeling A vertex-antimagic total labeling is a bijection from the set V E to the set of integers, {1, 2,..., V + E } where the weights of all the vertices are distinct with the additional restriction that all the weights form an arithmetic progression. Interestingly, to show a relation to magic labelings, we have the following result. Theorem 2.28 (Baca, Miller [2]). Every super-magic graph has a vertex-antimagic total labeling. To move back to the perspective of magic labeling, next I define the type of labeling that is the focal point of this paper. Distance magic labeling is the natural way to extend magic type labeling to the vertex set only. 8

13 2.9 Distance Magic Labeling Also known as sigma labeled or 1-vertex magic vertex labeled, a distance magic labeled graph is said to be distance magic. The labeling f is said to have a magic constant w. A graph G of order n has a distance magic labeling if there is a bijection f : V {1, 2,..., n } there is a positive integer w such that y N(x) f(y) = w for every x V where N(x) is the neighborhood set of vertex x [1]. It was shown by Miller, Rodger, and Simanjuntak that some common types of graphs are rarely distance magic. They proved the following. Theorem 2.29 (Miller, Rodger, Simanjuntak [23]). Some results. 1. P n is distance magic if and only if n {1, 3}. 2. C n is distance magic if and only if n = K n is distance magic if and only if n = W n is distance magic if and only if n = 4. Figure 4: Small Cases of Distance Magic [23] Much of the work in my project is inspired by the paper by S.B. Rao, T. Singh, and V. Parameswaran. They proved this as the main result of the paper. Theorem 2.30 (Rao, Singh, and Parameswaran [25]). The graph C n C k, n, k 3 is distance magic iff n = k and k 2 (mod 4). Of course, this is under the group, the integers. For the main results, I will be working with finite cyclic groups, but I will be using some of the ideas from their proofs to show my results [25]. Next, I will outline the proof given by Rao, Singh, and Parameswaran. They used a series of lemmas in order to eventually prove that C n C k, n, k 3 is distance magic iff n = k and k 2 (mod 4). The ideas from the first lemma are used later on to prove the new result. Here are the lemmas they used. Lemma 1. The graph C n C k, n, k 3 is not distance magic if n and k are both odd. Lemma 2. The graph C n C k, n k where n 0 (mod 4) is not distance magic. Lemma 3. C n C k, n, k 3 where n k and n 2 (mod 4) is not distance magic. Lemma 4. There is a general distance magic labeling for C n C k, n, k 3 where n = k and k 2 (mod 4) which shows sufficiency. 9

14 I note that the first three lemmas of the proof show the necessity of the conditions. The fourth lemma shows the sufficiency. That is, they provided a simple labeling under the conditions to finish the proof. Finally, at the end of their paper, they expanded by adding an additional theorem. Theorem 2.31 (Rao, Singh, and Parameswaran [25]). K m K n, m 2, n 3 is not distance magic. As a remark, we can note that K 2 K 2 is distance magic and is shown above as C 4 [25]. Looking at some other interesting results, Beena proved the following two theorems. Theorem 2.32 (Beena [3]). Given two positive integers m and n such that m n, the complete bipartite graph K m,n is distance magic if and only if the following are true. 1. m + n 0 or 3 (mod 4) and 2. either n (1 + 2m 1 2 or 2(2m + 2n + 1)2 = 1. Theorem 2.33 (Beena [3]). The product of paths P n P k is not distance magic for all n, k 3. Miller, Rodger, and Simanjuntak included in their paper some interesting negative results for labeling. Theorem 2.34 (Miller, Rodger, Simanjuntak [23]). If G contains two vertices u and v where N(u) N(v) = d(v) 1 = d(u) 1, then G is not distance magic. That is, if two vertices share all their neighbors except for one for each of them, then the graph is not distance magic. Theorem 2.35 (Miller, Rodger, Simanjuntak [23]). If G has n vertices with a maximum degree of and minimum degree of δ, then G does not have a labeling when ( + 1) > δ(2n δ + 1). Theorem 2.36 (Miller, Rodger, Simanjuntak [23]). Every k-regular graph with odd k does not have a magic labeling. These types of results can make it easy for future researchers to rule out many types of graphs as magic. Now that we have seen a number of different results about graph labeling including the products of graphs, one might ask if the product of cycles, other than the ones mentioned above, can ever be distance magic if we relax the definition somehow. If so, how can we work around the results by Rao, Singh, and Parameswaran? If not, what makes it impossible? I will demonstrate proofs for two new results that answer these questions, and they will take the form of a construction and a non-existence result, respectively. 10

15 3 MAIN RESULTS In an attempt to expand the work of Rao, Singh, and Parameswaran mentioned above, I introduce two new results with the perspective of finite groups. That is, consider a new type of labeling based on distance magic labeling. When the vertex weights are calculated under modular arithmetic, call this labeling group distance magic. The first result of this paper is to define a construction for a group distance magic labeling for C n C k, n 0 (mod 4) and k=4. Secondly, I show that when n and k are both odd there is not a group distance magic labeling under the group Z nk. Definition 3.1. A graph G of order n has a group distance magic labeling if there is a bijection f : V Z n and a positive integer w such that y N(x) f(y) = w for every x V where N(x) is the neighborhood set of vertex x, w Z n, and addition is in Z n. The first theorem will be proved by demonstrating a labeling for the graphs in question. In order to do this I will define a number of concepts and symbols as part of the first proof. Eventually, there will be a distinct magic constant for this type of graph. In the proof of the second theorem, I will show that the graphs in question are not group distance magic by demonstrating an algebraic contradiction. This will use a combination of graph theory and algebraic concepts to show that every graph of its type would have two vertices with the same label. Before constructing the labeling to prove Theorem 3.1, I will define some main concepts and general methods for working with products of cycles. I will show how to represent these graphs as a grid, how to differentiate notation for vertices, and how to calculate weights at vertices based on these representations. First, I will look at the grid representation. Given the torus nature of these graphs, the labelings can be easily represented using an n by k grid where boxes represent vertices, and vertices are adjacent if the boxes share a grid line. Since each vertex is adjacent to exactly four other vertices, consider boxes on the corners and sides to be adjacent to the boxes directly opposite them. The grid (Figure 6) represents the graph (Figure 5) where vertex a is adjacent to vertices b, c, d, and e. That is, the neighborhood set N(a) is made of b, c, d, and e. Figure 5: C 4 C 4 11

16 a e # c d # # # # # # # b # # # Figure 6: A C 4 C 4 grid representation Remembering that this graph has a torus structure, a different grid view of the same graph might look like Figure 7. # b # # c a e # # d # # # # # # Figure 7: A different view of C 4 C 4 Given this new representation for these graphs, denote the vertex labels of the graph as a i,j where j is given modulo 4. Figure 8 shows an example. a 0,0 a 0,1 a 0,2 a 0,3 a 1,0 a 1,1 a 1,2 a 1,3 a 2,0 a 2,1 a 2,2 a 2,3 a 3,0 a 3,1 a 3,2 a 3,3 a 4,0 a 4,1 a 4,2 a 4,3 a 5,0 a 5,1 a 5,2 a 5,3 a 6,0 a 6,1 a 6,2 a 6,3 a 7,0 a 7,1 a 7,2 a 7,3 Figure 8: C 8 C 4 with Indexing Define f(a i,j ) to be the label of a vertex a i,j, and define the neighborhood sum of a vertex a i,j by σ i,j where i is taken modulo n and j is taken modulo k. In this case, k = 4. As an example, Figure 9 shows σ 1,0 =( ) modulo 16, which equals 26 modulo 16. Finally, 26 is congruent to 10 (mod 16). 12

17 σ 1, Figure 9: C 4 C 4 Demonstrating Neighborhood Sums The method for labeling C n C k graphs is based on the property that neighborhood sums must be equal throughout. For example, σ 1,1 =σ 2,2. More generally, remembering that the indices are calculated (mod n) and (mod k), σ a,b =σ a+1,b+1. The direct consequence of this, for example, is that f(a 1,0 )+f(a 0,1 )=f(a 3,2 )+f(a 2,3 ). More generally, remembering that the indices are calculated (mod n) and (mod k), f(a c,d )+f(a d,c )=f(a c+2,d+2 )+f(a d+2,c+2 ). With these general definitions at hand, I can now demonstrate a construction of group distance magic labeling for this type of graph. Definition 3.2. Given a C n C 4 graph, the main superdiagonal is the set of vertices a i,j such that i = j (mod 4). The lth superdiagonal is the set of vertices a i,j such that i = j + l (mod 4). Theorem 3.1. A C n C 4 graph where n 0 (mod 4) has a group distance magic labeling. Proof. Label the sequence of vertices a i,j where i = j and j is taken modulo 4 by the following method. First, let a 0,0 = 0 and a 1,1 = 4. Next, for i even, let a i,j = (8 + a i 2,j 2 )(mod 4n). Similarly, for i odd, let a i,j = ( 8 + a i 2,j 2 )(mod 4n). Although the labeling is for a torus, for simplicity, think of the vertices a i,j where i = j as the main superdiagonal of the given grid. For k = 8, this part of the labeling is shown in Figure

18 0 # # # # 4 # # # # 8 # # # # # # # # 20 # # # # 24 # # # # 12 Figure 10: C 8 C 4 Example of Labeling Main Superdiagonal Skipping over a vertex, label the second superdiagonal (a 0,2, a 1,3, a 2,0,...) based on the already labeled vertices. Vertices a i,j where j = i + 2 and j is taken modulo 4 are labeled using a common sum. That is, a i,j + a i+1,j 1 = 4n 1. For example, when n = 8, the sum of the vertex (a i,j ) and its partner (a i+1,j 1 ) is 31. As a consequence, subtracting the given number (specifically, a label from the main superdiagonal) from 31 will give the desired label on the second superdiagonal. Labeling the remaining vertices is a matter of simply adding or subtracting 2 from certain neighbors. For the first superdiagonal (where j = i + 1 and j is taken modulo 4), a i,j = a i 1,j 2. For the third superdiagonal (where j = i + 3 and j is taken modulo 4), a i,j = a i 1,j + 2. The labeled graph where n = 8 is shown in Figure Figure 11: C 8 C 4 Example of Labeling To show that this labeling is group distance magic for all graphs of this type, consider the sum at an arbitrary vertex in each of the four diagonal cases below where i is 0, 1, 2, or 3. Also, for indexing simplicity, define m as a non-negative integer less than n 4. For each of these, when computed with modular arithmetic, there is a relatively simple formula (based on the labeling method) showing that all the sums are equal. For each of the four cases, the sum of its neighbors is shown in terms of the labeling and in reference to the given vertex. Again, 4n 1 is used to label the second superdiagonal as described above. It is used as a reference for two of the neighbors in each case. 14

19 1. The weight for each vertex on the main superdiagonal of the grid, σ 4m+i,i, is the sum of the following: (a) [4n 1 f(a 4m+i 1,i 1 ) 2] (above) (b) [f(a 4m+i,i ) + 2] (below) (c) [f(a 4m+i 1,i 1 ) + 2] (left) (d) [4n 1 f(a 4m+i,i ) 2] (right) Adding the four neighbors, we get [f(a 4m+i,i ) + 2] + [f(a 4m+i 1,i 1 ) + 2] + [4n 1 f(a 4m+i,i ) 2] + [4n 1 f(a 4m+i 1,i 1 ) 2] = 8n 2 (mod 4n) = w 2. The weight for each vertex on the first superdiagonal, σ 4m+i,i+1 is the sum of the following: (a) [f(a 4m+i,i+1 ) + 2] (above) (b) [4n 1 f(a 4m+i,i+1 ) 2] (left) (c) [4n 1 f(a 4m+i+1,i+1+1 ) 2] (below) (d) [f(a 4m+i+1,i+1+1 ) + 2] (right) Adding the four neighbors, we get [f(a 4m+i,i+1 )+2]+[f(a 4m+i+1,i+1+1 )+2]+[4n 1 f(a 4m+i,i+1 ) 2]+[4n 1 f(a 4m+i+1,i+1+1 ) 2] = 8n 2 (mod 4n) = w 3. The weight for each vertex on the second superdiagonal, σ 4m+i,i+2 is the sum of the following: (a) [4n 1 f(a 4m+i 1,i+2 1 ) ] (above) (b) [f(a 4m+i 1,i+2 1 ) 2] (left) (c) [f(a 4m+i,i+2 ) 2] (below) (d) [4n 1 f(a 4m+i,i+2 ) + 2 8] (right) Adding the four neighbors, we get [f(a 4m+i,i+2 ) 2]+[f(a 4m+i 1,i+1 ) 2]+[4n 1 f(a 4m+i,i+2 )+2 8]+[4n 1 f(a 4m+i 1,i+1 ) ] = 8n 2 (mod 4n) = w 4. The weight for each vertex on the third superdiagonal, σ 4m+i,i+3 is the sum of the following: (a) [f(a 4m+i,i+3 ) 2] (above) (b) [4n 1 f(a 4m+i,i+3 ) 8 + 2] (left) (c) [4n 1 f(a 4m+i+1,i ) ] (below) (d) [f(a 4m+i+1,i ) 2] (right) Adding the four neighbors, we get [f(a 4m+i,i+3 ) 2]+[f(a 4m+i+1,i ) 2]+[4n 1 f(a 4m+i,i+3 ) 8+2]+[4n 1 f(a 4m+i+1,i )+8+2] = 8n 2 (mod 4n) = w 15

20 That is, all vertices have a weight of (8n 2) modulo 4n which is congruent to (4n 2) modulo 4n under the given labeling. This means there is a group distance magic labeling for each graph of this type. Next, one might wonder what happens when the cycles have an odd number of vertices. Interestingly, changing to this gives a quite different result. Before introducing my second result, I must prove a proposition for later use. Proposition 3.2. The mapping φ(x) = 2x from Z 2n+1 to Z 2n+1 is a bijection. elements a and b, 2a = 2b implies a = b under the mapping. As a result, for Proof. Since a mapping from Z 2n+1 to Z 2n+1 is a bijection if and only if it is a surjection, I will prove only the surjective property. Notice that for any element x Z 2n+1, either x = 2z and 0 z n or x = 2z + 1 and 0 z n 1. I will show for all x in Z 2n+1 there exists a y such that φ(y) = 2y = x. To do this, I will use two cases. For the case when x = 2z and 0 z n, y = z is the preimage since φ(y) = 2y = 2z = x. For the case when x = 2z + 1 and 0 z n 1, y = n + z + 1 is the preimage since φ(y) = 2y = 2(n + z + 1) = (2n + 1) + (2z + 1) = 2z + 1 = x. Thus, φ(x) is a bijection. For the purposes of this paper, 2x = 2y x = y. Now that I have shown this injective property, I can prove my second result. The following is my second theorem based on the work by Rao, Singh, and Parameswaran [25]. Theorem 3.3. A C n C k graph where n and k are both odd is not group distance magic. Proof. I will use the method from the proof of Lemma 1 in [25] to show these graphs are not group distance magic. Considering the neighborhood sums of a 1,1 and a 2,2, as was described above, Using the same logic, f(a 1,0 )+f(a 0,1 )=f(a 2,3 )+f(a 3,2 ). f(a 2,3 )+f(a 3,2 )=f(a 4,5 )+f(a 5,4 ). These equalities continue such that for some integer x the sum of each pair in the pattern is equal to the sum f(a 2x+1,2x+0 )+f(a 2x+0,2x+1 ) when calculating the indices under the appropriate finite group. In other words, thinking of the graph as a torus, the pairs of equal sums continue to wrap around. To show there exists an x such that the sum f(a 2,1 )+f(a 1,2 ) is included, we have the congruences, 2x (mod n) 2x (mod k) 2x (mod n) 2x (mod k) which reduce to 2x 1 (mod n) 2x 1 (mod k). 16

21 This implies there exist positive integers s and t such that 2x=nt + 1=ks + 1 nt=ks. Let d=gcd(n, k). It follows that the equation is satisfied by t= k d and s= n nk d. Substituting, 2x= d +1. Now, n, k, and d are all odd integers; therefore, x= 1 2 ( nk d + 1) exists as an integer to satisfy f(a 2,1)+f(a 1,2 ). Next, by the same logic, it can be shown that the analogous is true for sums along perpendicular diagonals of the torus graph. That is, repeating the proof starting with the neighborhood sums of a 1,1 and a 2,0, there exists an x such that f(a 0,1 )+f(a 1,2 )=f(a 2x+0,2x+1 )+f(a 2x+1,2x+2 )= f(a 1,0 )+f(a 2,1 ). To summarize, it has now been shown that and f(a 1,0 ) + f(a 0,1 ) = f(a 2,1 ) + f(a 1,2 ) f(a 0,1 ) + f(a 1,2 ) = f(a 1,0 ) + f(a 2,1 ). Visually, for C 7 C 5 the labels on the grid are shown in Figure 12. # a 0,1 # # # a 1,0 # a 1,2 # # # a 2,1 # # # # # # # # # # # # # # # # # # # # # # # Figure 12: C 7 C 5 Demonstrating Layout Remembering that the labels are integers, we can have a system of equations and solve. Subtracting gives f(a 1,0 )+f(a 0,1 )=f(a 2,1 )+f(a 1,2 ) f(a 0,1 )+f(a 1,2 )=f(a 1,0 )+f(a 2,1 ) Solving the new equation, we have Applying Proposition 3.2, f(a 1,0 ) f(a 1,2 )=f(a 1,2 ) f(a 1,0 ). 2f(a 1,0 )=2f(a 1,2 ). f(a 1,0 )=f(a 1,2 ). In words, two vertices must have the same label. This equality contradicts the assumption that there is a bijection from the labeling to the integers modulo nk. As a result, there is no labeling for C n C k while both n and k are odd. 17

22 4 CONCLUSION We have seen that there are many types of graph labelings with highly varied results. Small changes can have a substantial effect on whether or not a graph can be labeled in a certain way. In this case, by introducing a new magic type labeling as a variation on an established one, I was able to label a type of graph that was not possible before the change. At the same time, I showed that the result did not change for odd cycle graphs when using group distance magic. Stated again, there exists a group distance magic labeling for C n C k graphs where k = 4 and n 0 (mod 4). Moreover, the magic constant w equals 4n 2. When n and k are both odd, C n C k is not group distance magic. These results open up new questions for future research. The next logical area to explore would be the case of C n C k where both n and k are equivalent to 0 modulo 4. After that, one would wonder about the cases where n and k are different combinations of 0 modulo 4 and 2 modulo 4. That is, the ideas to explore are the cases not covered by the negative result. 18

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