Vertex-Magic. Daisy Cunningham Advisor : Dr. Crista Coles Elon University Elon, NC

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1 Vertex-Magic Daisy Cunningham Adisor : Dr. Crista Coles Elon Uniersity Elon, NC dsc88@aol.com Introduction Graph theory was introduced by Leonhard Euler in when he began discussing whether or not it was possible to cross all of the bridges in the city of Kaliningrad, Russia only once. The publication of the problem and his proposed solution began what is now known as graph theory. Since then, the topic has been thoroughly studied by mathematicians such as Thomas Pennyngton Kirkman and William Rowan Hamilton. In Euler s original problem, ertices represented locations around the city which were connected by bridges, or the edges of the graph. For more information on the history of graph theory, see [] and []. One of the most famous problems in graph theory is the Four Color Conjecture proposed by Francis Guthrie, a student of Augustus DeMorgan, around 80 []. The main objectie of this problem is to color a map in such a way that no two adjacent countries, those sharing a border, are the same color. In this problem, countries are represented with ertices. If two countries share a border, then their corresponding ertices are adjacent. The conclusion showed that any map on a sphere can be colored with four colors. The ertices of a graph can be labeled in many different ways. Another way to label ertices is with numbers. An interesting ertex labeling with numbers is ertex-magic. Vertex-magic graphs are graphs labeled with numbers in which eery ertex and its incident edges add up to the same number. This number is called the magic number. It is still unknown what types of graphs are ertexmagic and which are not. One type of graph that has interesting ertex-magic properties is the cycle graph. The following questions hae interesting solutions regarding the labelings of ertex-magic cycle graphs: Can sharp bounds be found for the magic number of a gien cycle graph? What are some properties of ertex-magic cycle graphs with both an een and odd number of ertices?

2 Preliminary Information Before looking at ertex labelings, we must first look at some basic concepts of graph theory. Definition.. A graph is a finite set of ertices and edges where eery edge connects two ertices. Any two ertices that share an edge are said to be adjacent. Any two edges that are incident to the same ertex are said to be adjacent. If an edge connects two ertices, then that edge is considered incident to both of these ertices. Definition.. A graph is connected if any two ertices are connected by a sequence of adjacent edges. One unique type of graph that has ery interesting ertex-magic characteristics is the cycle graph. Definition.. A cycle graph is a connected graph where eery ertex is adjacent to two other distinct ertices. Figure : Three examples of cycle graphs. Definition.. If a graph G with ertices and e edges is labeled with numbers through + e such that eery ertex and its incident edges adds up to the same number, then G is a ertex-magic graph. We call this number the magic number. Definition.. If a graph G with ertices and e edges is labeled with numbers through + e such that eery edge and its two adjacent ertices adds up to the same magic number, then G is an edge-magic graph. In Figure, we hae an example of ertex magic graphs. In the left graph, notice that each ertex and its incident edges add up to 0. This graph is also edge-magic because each edge and its adjacent ertices adds up to. Eery cycle graph that is ertex-magic with a magic number, k, can also be labeled so that it is edge-magic with a magic number k. By maintaining the order of the ertex and edge labelings and rotating them clockwise, an edge-magic cycle graph can be created from a ertex-magic cycle graph. Figure shows just one way to create a ertex-magic graph with three ertices. Depending upon the number of ertices and edges, a graph can be labeled in different ways with different magic numbers.

3 Figure : A cycle graph with a magic number of 0. If the labels are rotated clockwise, an edge-magic graph is created with a magic number of 0. A Lower Bound for the Magic Number In order to find a ertex-magic cycle graph, one cannot possibly try eery number as the magic number. Haing a range for the magic number would be helpful in finding ertex-magic labelings of a graph. One can determine a range for the magic number based on the number of ertices and edges of the graph. Lemma.. If G is a ertex-magic graph with ertices and e edges, then ( + e)( + e + ) + E sum = k. Proof. Let V sum be the sum of all ertex labels and E sum be the sum of all edge labels of graph G. Since each edge is incident to two ertices, each edge label will be counted towards its two adjacent ertices magic numbers. Therefore, V sum + E sum = k. Since edges and ertices are labeled through + e, Therefore, V sum + E sum = ( + e)= ( + e)( + e + ) + E sum = k, ( + e)( + e + ) + E sum = k. ( + e)( + e + ). Theorem.. Let G be a graph with ertices and e edges. If G is a ertexmagic graph, then the magic number, k, is bounded such that e(e + ) + ( + e + )( + e) k e + e(e + ) + ( + e + )( + e).

4 Proof. From Lemma., we can conclude that ( + e + )( + e) E sum = k. The minimum E sum occurs when numbers through e are assigned to the edges. Therefore, e(e + ) E sum. The maximum E sum occurs when numbers + through + e are assigned to the edges. Hence, Hence, E sum = e + i, i= e + i= = e + e i, i= e(e + ). e(e + ) e(e + ) E sum e +, e(e + ) ( + e + )( + e) e(e + ) k e +, e(e + ) + ( + e + )( + e) k e + e(e + ) + ( + e + )( + e). In this paper, we will be looking at ertex-magic labelings of cycle graphs. We can now use Theorem. in order to find a range for the magic number for all cycle graphs. Corollary.. Let G be a cycle graph with ertices. If G is a ertex-magic graph, then the magic number, k, is bounded such that and + k where is odd, + k where is een. Proof. For eery cycle graph, = e. By substituting in for e in Theorem., we obtain,

5 ( + ) + ( + )() k, = + + +, = +. When is odd, + is diisible by, so the minimum magic number for a cycle graph with an odd number of ertices is +. For an example of how to apply this bound to a ertex-magic problem, see example.. The formula, k = + + +, is possible only if is odd, and a better bound can be found if is een. The minimum ertex labeling for a cycle graph occurs when numbers through are assigned to the edges. From Lemma., ++ E sum = k. This implies that E sum must be diisible by. What will the edge labelings be in order to hae a minimum ertex labeling for an een cycle graph where E sum is diisible by? If is een, can edges be labeled with numbers through? If this labeling is used, then k = We know that + must be diisible by, howeer if is een, then + cannot be diisible by. Therefore, the numbers through cannot be used for the edges to find a minimum labeling. Although the edge labeling of through cannot be used for a minimum labeling, consider adding to one of the labels. See examples. and. in order to see how E sum can be adjusted so that the magic number is an integer. Adding to E sum if is een will create a alid edge labeling. Note that this new E sum is diisible by. E sum = , As before, = ( + ) +.

6 k = + + E sum, + + ( + ) + = + + +, = +., Example.. Let G be a graph with = ertices. If G is ertex-magic what would the minimum magic number be? The smallest E sum comes from labeling the edges with numbers through. Referring to Corollary., the magic number for this particular graph would be () + =. Example.. Let G be a cycle graph with = ertices. What are some combinations of ertex labelings that could be used in creating a ertex-magic graph with a minimum k? Begin by making E sum = Since =, and does not diide E sum = 0, then E sum must be adjusted such that E sum is diisible by. The edge labels cannot be shifted down because the smallest numbers are already on the edges. The closest multiple of after 0 is. Therefore, E sum must be shifted from 0 to by adding. There are numerous ways of doing this. By adding to, E sum becomes Since the sum of the edges equals, {,,, } is a alid edge labeling for finding a possible minimum k. Example.. Let G be a cycle graph with = ertices. What is a alid E sum that can be used to make G ertex-magic with a minimum k? By using the same technique, the edge labelings can be altered for =. Let E sum = Since this sum of is not diisible by, we must add to obtain which is diisible by. Therefore, the minimum magic number is when =. Knowing whether a graph has an een or odd number of ertices allows us to know the minimum bound on the magic number. It also determines what the possible edge labelings are in order to create a minimum ertex-magic graph. An Upper Bound for the Magic Number We hae just seen how to determine the lower bound of the magic number of any cycle graph. There also exists an upper bound for any ertex-magic cycle graph. Again, the bounds depend on whether the graph has an een or odd number of ertices.

7 Corollary.. Let G be a graph with ertices. If G is a ertex-magic graph, then the magic number, k, is bounded such that k + where is odd, and k + where is een. Proof. The upper bound would occur on any graph when the largest numbers, + through, are placed on the edges. Therefore, E sum = + i, i= = ( + ) +, ( + ) =. Also, k = + + E sum, + + ( + ), = + + +, = +. As in the case of the lower bound, must diide + and must diide E sum. For any cycle graph with an odd number of ertices, this holds true when the edges are the largest numbers. So, the maximum possible k for any cycle graph with an odd number of ertices is +. Howeer, a cycle graph with an een number of ertices is more difficult. When looking at the upper bound for k of a cycle graph with an een number of ertices, assume that the numbers + through are placed on the edges. Since + is not diisible by, one must alter the edge labelings. To find the lower bound for the magic number when is een, the number E sum was increased, but in this case, the numbers cannot be increased because no label can be greater than. So, in order to find an E sum diisible by, the original E sum must be decreased, see examples. and..

8 For any cycle graph with an een number of ertices, the number must be subtracted from the original sum. Therefore, Thus, ( + ) E sum = = ()., k = + + E sum, + +, = +. Example.. Let G be a cycle graph with = ertices. What are some possible combinations of ertex labelings such that G has a maximum k? Label the graph such that E sum = 8+++ =. The largest number less than that is diisible by is. Therefore, or, must be subtracted from E sum making the magic number. One possible edge labeling is {8,,,}. By changing the to a, E sum becomes 8+++ which is diisible by. Therefore, the edge labelings could possibly be used to label this graph with the maximum k. Example.. Let G be a graph where =. What is the maximum E sum for G such that G is ertex-magic? If G is labeled with the largest numbers on the edges, G has an E sum of. The next E sum less than that diides is 8. Thus, must be subtracted from the original E sum to make the new E sum equal to 8. This edge labeling has a magic number of. Knowledge of the number of ertices on a cycle graph determines the upper and lower bounds for the magic number of that graph. These bounds are ery useful when trying to create a ertex-magic cycle graph. By knowing the upper and lower bounds, we can conclude that if a graph has an odd number of ertices, then there exist + possible magic numbers. If a graph has an een number of ertices, then there exist possible magic numbers. 8

9 Maximum and Minimum Magic Number for Odd Cycle Graphs Now that we hae seen bounds for the magic number, let us look at some concrete algorithms for ertex-magic labelings. Labeling the edges with numbers + through will produce a ertex-magic graph with the maximum magic number for odd cycle graphs. On the other hand, assigning the numbers through to the edges and the numbers + through to the ertices, will produce a ertex-magic graph with the minimum magic number. Theorem.. Let G be a cycle graph with ertices where is odd. There exists a ertex-magic labeling with the numbers to located on the ertices and a magic number of +, the upper bound for the magic number. (/)+(/) - (/)-(/) (/)-(/) - Figure : Vertices of a general cycle graph labeled with the smallest numbers. Proof. Label the ertices with the consecutie numbers through in a clockwise manner. We will assume in this paper that no two edges of the cycle graph cross. An edge is to the right of a ertex if it is adjacent and clockwise to that ertex, and an edge is to the left of a ertex if it is adjacent and counterclockwise to that ertex. Starting with the edge to the right of the ertex labeled, go clockwise around the polygon twice labeling eery other edge with consecutie numbers + through in descending order starting with (Figure ). Any ertex of a cycle graph can be categorized into one of the following two categories:

10 Description of Case Vertex Label Left Edge Label Right Edge Label Eery other ertex i + starting with i = 0,..., + i i Eery other ertex i + starting with i = 0,..., Table i i A ertex in the first case will hae a magic number of i i + i = +. A ertex in the second case will hae a magic number of i + + i + i = +. Thus, this graph is ertex-magic with a magic number of +. Figures and are examples of ertex-magic labeling with the smallest numbers on the ertices. Figure : A cycle graph with ertices and a magic number of. 8 0 Figure : A cycle graph with ertices and a magic number of. Now, assign the largest numbers to the ertices. As expected, the maximum magic number is fixed and depends only on the number of ertices. 0

11 Theorem.. Let G be a cycle graph with ertices where is odd. There exists a ertex-magic labeling with the numbers + to located on the ertices and a magic number of +, the lower bound for the magic number. (/)+(/) (/)-(/) - + (/)-(/) Figure : The ertices of a cycle graph labeled with the largest numbers. Proof. Using the same ideas as in Theorem., a cycle graph can be labeled with the largest numbers on the ertices. Label the ertices with the consecutie numbers + through in a clockwise manner. Starting with the edge to the right of ertex +, go clockwise around the polygon twice labeling eery other edge with consecutie numbers through in descending order starting with (Figure ). The ertices of the cycle graph can be categorized into one of the following two categories: Description of Case Vertex Label Left Edge Label Right Edge Label Eery other ertex + + i starting with + i = 0,..., + i i Eery other ertex + + i starting with + i = 0,..., Table i i A ertex in the first case will hae a magic number of + + i + i + + i = +. A ertex in the second case will hae a magic number of

12 + + i + i + i = +. This cycle graph is ertex-magic and has a magic number of +. Some examples of this type of labeling are shown in Figures and 8. Figure : A cycle graph with ertices and a magic number of. 0 8 Figure 8: A cycle graph with ertices and a magic number of. We hae created algorithms for producing ertex-magic odd cycle graphs with minimum and maximum magic numbers. Since there exist odd cycle graphs where the magic number is equal to the bounds, we conclude that the bounds for odd cycle graphs are sharp. The bounds for een cycle graphs are conjectured to be sharp, howeer finding an algorithm that makes these bounds sharp is more difficult. Minimum Magic Number for Een Cycle Graphs A ertex-magic cycle graph with an een number of ertices can be found for the minimum k. Wallis, in [], gies the following construction for a ertex-labeling for een cycle graphs that gies the minimum magic number. Conjecture.. Let G be a cycle graph with ertices where is een. There exists a ertex-magic labeling for G with the minimum magic number k = +. If we can find edge labelings that create a ertex-magic graph, then by adding the two incident edges of a ertex and subtracting that sum from the magic number, the ertex labels can be easily obtained. The edges of a cycle graph with an een number of ertices can be labeled as follows:

13 e e e e e Figure : The edges of a cycle graph. Let = n. The alue of n can be either een or odd, and the construction of the ertex-magic graph depends on n. If n is een, the following is a construction for how to label the edges: i +, i =,,..., n +, n, i =, n + i e i =, i =,,..., n, n + i, i = n +, n +,..., n, i +, i = n +, n +,..., n. Figures 0 and are two examples of this type of labeling. A cycle graph with = 8 and n = should hae a minimum k of. By using the gien edge labelings for an een n, one can find the edges and ertices for this graph. 0 8 Figure 0: A cycle graph with the minimum magic number of. Another example of this type of labeling occurs when n =, or =. Using Conjecture., we calculate that the magic number should be.

14 Figure : A cycle graph with the minimum magic number of. If n is odd, a similar technique can be used to label the cycle graph. i +, i =,,..., n, n, i =, n + i +, i =,,..., n, n + e i =, i = n +, n + i, i = n +, n +..., n, i +, i = n +, n +,..., n, n +, i = n. Figures and show ertex-magic cycles where n is odd. If =, then n =, and the magic number is. 8 0 Figure : A cycle graph with the minimum magic number of.

15 The following graph shows a ertex-magic labeling when = 0. n =, and the magic number is. Thus, Figure : A cycle graph with the minimum magic number of. We hae just shown how to produce ertex-magic graphs with the minimum magic number for both een and odd cycles. We hae also shown how to create ertex-magic graphs with the maximum magic number for odd cycles. Maximum Magic Number for Een Cycle Graphs Using the same logic as in Section, one can find an algorithm for creating a ertex-magic cycle graph with the maximum magic number. Theorem.. Let G be a cycle graph with ertices where is een. If Conjecture. holds, then there exists a ertex-magic labeling for G with the maximum magic number k = +. One can see that in order to obtain the minimum k in Conjecture. for an een, the edge labeling should be,,,...,, +. In order to obtain the maximum k, the edge labels should be,,..., +, +. These alues are exactly + minus the numbers on the edges in the minimum labeling. The following cases produce the edge labelings for a maximum magic ertex graph. Using the maximum k, we can subtract the sum of the two edges in order to label each ertex. Again, let = n and let n be een.

16 i + +, i =,,..., n +, n +, i =, e i = n + i n + i +, i =,,..., n, +, i = n +, n +,..., n, i + +, i = n +, n +,..., n. Figures and show ertex-magic cycle graphs where n is een. 8 0 Figure : A cycle graph with the maximum magic number of Figure : A cycle graph with the maximum magic number of. Now, let n be odd. Subtracting the edge labels of the algorithm in Conjecture. from and adding, a new edge labeling with a maximum k can be obtained.

17 i + +, i =,,..., n, n +, i =, n + i + +, i =,,..., n, e i = n + +, i = n +, n + i +, i = n +, n +..., n, i + +, i = n +, n +,..., n, (n + ) +, i = n. Figures and are examples of cycle graphs when n is odd. 8 0 Figure : A cycle graph with the maximum magic number of Figure : A cycle graph with the maximum magic number of. Different algorithms can be found for other odd cycle graphs such that the magic numbers are within the range for k. Since we hae found algorithms for

18 creating ertex-magic graphs such that the magic number is the minimum and maximum bound, we can conclude that the bounds are sharp. 8 Odd and Een Numbers on the Vertices Another possible labeling for odd ertex magic cycle graphs is to assign the odd numbers to the ertices. Theorem 8.. Let G be a cycle graph with ertices where is odd. There exists a ertex-magic labeling for G with the odd numbers from to located on the ertices and a magic number of Figure 8: The ertices of a general cycle graph labeled with odd numbers. Proof. In order to label an odd cycle graph with odd numbers on the ertices, the ertices can be labeled with the consecutie odd numbers through in a clockwise manner. Starting with the edge to the right of the ertex labeled, go clockwise around the polygon twice labeling eery other edge with consecutie een numbers through in descending order starting with (Figure 8). In general, any ertex of this labeling can be categorized into one of the following two categories: Description of Case Vertex Label Left Edge Label Right Edge Label Eery other ertex i + starting with i = 0,..., + i i Eery other ertex i + starting with i = 0,..., i i Table 8

19 In the first case, the magic number is i i + i, or +. In the second case, the magic number is i++ i+ i, which is also +. Therefore, the graph is ertex-magic with a magic number of +. Some examples of such labelings and their magic numbers are gien in Figures and 0. Figure : A cycle graph with ertices and a magic number of. 0 8 Figure 0: A cycle graph with ertices and a magic number of. Using similar ideas, one can label an odd cycle graph with the een numbers on the ertices. Like the odd numbers on the ertices, the een numbers on the ertices produce a pattern that is dependent upon the number of ertices. Theorem 8.. Let G be a cycle graph with ertices where is odd. There exists a ertex-magic labeling for G with the een numbers from to located on the ertices and a magic number of +. Proof. A similar technique to the preious proof can be used in order to label a cycle graph with een numbers on the ertices. The ertices are labeled with the consecutie een numbers through in a clockwise manner. Starting with the edge to the right of the ertex labeled, go clockwise around the polygon twice labeling eery other edge with consecutie odd numbers through in descending order starting with (Figure ). For this labeling, a ertex can be categorized into one of the following two categories:

20 Figure : The ertices of a general cycle graph labeled with een numbers. Description of Case Vertex Label Left Edge Label Right Edge Label Eery other ertex i + starting with i = 0,..., i i Eery other ertex i + starting with i = 0,..., i i Table In the first case, the magic number is i + + i + i, or +. In the second case, the magic number is i + + i + i, or +. Therefore, the graph is ertex-magic with a magic number of +. Some examples of this type of labeling are gien in Figures and. Figure : A cycle graph with ertices and a magic number of 0. 0

21 0 8 Figure : A cycle graph with ertices and a magic number of. Finding the aerage of the minimum and maximum bounds, we get, = +. This alue is not a possible alue for k. Howeer the two middle k alues, + = + and + = + occur when the een and odd numbers are placed on the ertices respectiely. Conclusion This paper addresses labeling graphs in such a way that they are ertex-magic. Exploring bounds for magic numbers is an interesting problem for all graphs. We are able to obtain more accurate bounds if we limit the graphs to be cycle graphs. If a graph has an odd number of ertices, algorithms can be easily found to produce different magic-ertex graphs with the maximum and minimum magic number. Also, eery cycle graph with an odd number of ertices can be made into a ertex-magic graph if the odd numbers or een numbers are placed on the ertices. Some interesting problems arise when one begins to look at cycle graphs with an een number of ertices. Bounds for the magic number change, and it becomes harder to make these graphs ertex-magic. We hae shown some algorithms for finding ertex-magic cycle graphs with a magic number that lies within the bounds. In [], Wallis explores a ariety of properties of magic graphs, including some of the properties discussed aboe. Vertex-magic graphs continue to attract the attention of many graph theorists. Some interesting questions that arise from this paper are: For a gien cycle graph, is there a ertex-magic labeling associated with eery magic number within the bounds? Can an algorithm be created for finding all possible ertexmagic labelings for a gien cycle graph? Can these properties be generalized for any graph? References [] Bollobas, Bela. Graph Theory: An Introductory Course, Springer-Verlag, New York, NY,.

22 [] Finkbeiner, D. and Lindstrom, W. A Primer of Discrete Mathematics, W.H. Freeman and Co., New York, NY, 8. [] Locke, Stephen C. Graph Theory. Online Posting. [] Wallis, W.D., Magic Graphs, Birkhauser, New York, NY, 00.