#### Enumerating Of Star-Magic Coverings And Critical Sets

Volume 4 - Issue 5, May 2020 Edition

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Author(s)

Anisatul Farida, Ratna Puspita Indah

Keywords

, critical sets, complete bipartite graph

Abstract

A simple graph admits a H-magic covering if every edge in G belongs to a subgraph of G that isomorphic to H. Let be a complete bipartite graph on m and n vertices. Then graph G is a if there is a total labeling such that for each subgraph of G isomorphic to , there exists , where m(f) is a positif integer called magic sum. When f(V)= we say that G is and we donete its supermagic-sum by s(f). The aim of this research is to list all possible H-magic covering on complete bipartite graph, where H is star magic, especially for n=3. After we list all possible star magic covering on complete biparite graph, we tke s one example of the labeling to find the critical sets. The method f this research is a literary study and we carried outthe enumertion of all possible existing labeling. This research get a sum of 836 labeling which are not isomorphic. After we get list of all possible labeling, we choose a set of label vertices {1,3,4,5,7,10} from which we establish some 28 critical set, 21 of them are of size 1 and the others are of size 2

References

[1]. Baskoro, E.T, Critical set in edge-magic total labelings, J. Combin. Math Combin. Comput. 55(2005), 32-42.

[2]. Bondy, J.A. and U.S.R Murty, Graph Theory with Applications, The Macmilliarn Press Ltd, London, 1976.

[3]. Gallian, J.A, Dynamics Survey of Graph Labeling, Electron, J. Combi 14 (2010)#DS6

[4]. Gutierrez, A. and A. Blado, Magic Coverings J. Combin. Comput, 55 (2005), 43-56.

[5]. Blado , A. and J. Morages, Cycle-Magic Graphs, Discrete Mathematics. 23. (2007)

[6]. Maryati, T.K.E.T. Baskoro and A.N.M Salman, Ph-Super magic Labelings of some Trees, J. Combin. Math Combin.Comput(2008), 197-204

[7]. Ngurah, A.A.G.A.N.M Salman, and L. Susilowati, H-supermagic Labelings of Graphs, Discrete Mathematics (2010), 1293-1300.

[8]. Rosa, A, On Certain Valuation of the Vertices of a graph, (International Symposium, Rome, July 1966)(1967), 349-355

[9]. Rosa, A. and A. Kotzig, Magic Valuations of finite graphs, Canad, Math. Bull (1970), 451-456

[10]. Sedlacek, J, Theory of Graphs and its Applications , House Szechoslavak Acad. Sci, Prague (1964), 163-164

[11]. Wallis, W.D, Magic Graph, Birkhauser, Boston, Basel, Berlin. 2001.