P Selvagopal (ps_gopaal@yahoo.co.in) |

Mathematics, Manonmaniam Sundaranar University, Tirunelveli |

October, 2009 |

## Abstract |

A total labeling is a function from the set of vertices union the set of edges of a (p,q) graph G to the set of natural numbers {1,2,â€¦, p+q}. A total labeling f is said to be magic if for all edges xy, f(x)+f(y)+f(xy) are the same. An edge-covering of a graph G is a family of different subgraphs H1, H2, H3, â€¦, Hk of G such that every edge of E belongs to at least one of the subgraphs Hi for 1 ï‚£ i ï‚£ k. If every Hi is isomorphic to a given graph H, then we say that G admits an H âˆ’ covering. A graph G = (V, E) that admits an H-covering is called H âˆ’ magic if there exists a total labeling such that for all subgraphs of G isomorphic to H, we have, the sum of vertex labeling and edge labeling are the same. And is said to be H-supermagic if the vertices of G receive the smaller labels {1, 2, â€¦,p} . The constant value that every copy of H takes under the labeling f is denoted by m( f ) in the magic case and by s( f ) in the supermagic case. In this thesis we investigate some graphs that admit cycle-coverings, construct some H-supermagic graphs from a given graph H. We also introduce the notion of H-supermagic dual of a H-suermagic labeling and H-supermagic strength of graphs admitting H-supermagic labeling. We investigate the generalised prism Cm Ã— Pn, m > 4 , the ladder graph P2 Ã— Pn , the grid P3 Ã— Pn , the generalised antiprism , the triangular ladder Ln, the Fan Graph Fn and k-polygonal snake and proved they are cycle-supermagic. We construct a chain of any 2-connected graph H, garland graph and linear garland of a 2-connected graph H and prove they are H-supermagic. Also we prove that the edge amalgamation of a finite number of graphs isomorphic to a given 2-connected graph H, and the one point union of a finite number of copies of a 2-connected graphs are H-supermagic. We establish the relation between H-supermagic labeling and its dual. Also we find the H-supermagic strength of some graphs. As a special case we find the bounds for the Ph-supermagic strength of path graphs. |