Signed Edge Domination Numbers in Complete Tripartite Graphs

1. Signed Edge Domination Numbers in Complete Tripartite Graphs Dana Collins, Tessa Gromoll. and Daniel Hartman Advisor: Dr. Abdollah Khodkar Department of Mathematics, The University of West Georgia UWise (University of West Georgia Institutional STEM Excellence) Student Research Program Abstract  A complete tripartite graph is a graph with three sets of vertices, such that the vertices in each set are connected to all of the other vertices in all of the other sets by edges. In addition, no two vertices in the same set are connected. The closed neighborhood of an edge is all of the edges which have one vertex in common with that edge. The weight of a subset of edges is the sum of all of the labels of all of the edges in that subset. A labeling of edges such that the weight of each neighborhood of each edge in the graph is greater than or equal to one is called a signed edge dominating function. The weight of a signed edge dominating function of a graph is the sum of all of the labels of all the edges of the graph. In this project, we investigate how to find the minimum weight of a signed edge dominating function of a tripartite graph; which is called the signed edge domination number of that graph. In order to carry out this research we have explored graphic sequences as a tool for counting such weights. As a result of this research, we have generalized several lemma’s regarding the signed edge domination numbers in complete tripartite graphs. Introduction Methods Results Future Studies Consider Lemma 1, where m, n, and p are three positive even integers such that m + n ≤ p. Below is an example of a complete tripartite graph K2,2,4 , where m = 2, n = 2, and p = 4. The red edges represent a -1 labeling, and the blue edges represent a +1 labeling. The weight of each vertex is the sum of the labeling of the connected edges. One way to compute the weight of the graph is to add up the weight of the vertices and divide that number by two. Notice the weight of this graph is 4, which is the minimal weight for a K2,2,4 graph; also known as the signed edge domination number of the graph (w(f)). • There are many ways to continue this research. Some interesting topics include: • Further research can be carried out on cases where m+n ≥ p. • Also generalizing signed edge domination numbers for complete n-partite graphs. A complete graph is a graph in which each vertex in each set is connected to every other vertex in all of the other sets by an edge. A tripartite graph is a graph with three sets of vertices. Each edge on the graph can be labeled either as either a positive or negative weight of one. The weight of a vertex on a graph is the sum of all of the weights of the edges connected to that vertex. The weight of the entire graph is the sum of all of the weights of the edges. The purpose of this research is to generalize the optimal labeling (known as the signed edge dominating function) of each edge, such that the weight of the graph is at its minimal (known as the signed edge domination number) and the closed neighborhood of every edge on a graph must be greater than or equal to one. For our research we let m, n, and p be the number of vertices in each set. The graph denoted as Km,n,p where m ≤ n ≤ p, and m+n ≤ p. To carry out our research, we examined cases where m, n, and p can be even or odd integers. References • S. Akbari, S. Bolouki, P. Hatamib and M. Siami, On the signed edge domination number of graphs, Discrete Mathematics 309 (2009), 587-594. • A. Carney and A. Khodkar, Signed edge k-domination numbers in graphs, Bulletin of the Institute of Combinatorics and its Applications 62 (2011), 66-78. • B. Xu, On signed edge domination numbers of graphs, Discrete Mathematics 239 (2001), 179-189. Consider Lemma 2, where m, n, p are three positive odd integers such that m + n ≤ p. Below is an example of a complete tripartite graph K3,3,7, where m = 3, n = 3, and p = 7. Notice the weight of this graph is 3+3+1 =7, which is the minimal weight for a K3,3,7 graph. Acknowledgements • We would like to acknowledge the following for their help in making this research possible: • • Dr. Abdollah Khodkar • Dr. S. Swamy Mruthinti • Dr. Scott Sykes • This research was made possible by the funding of the UWISE Program