Maximum Hexagon Packing And Generalized Almost Resolvable26-cycle Systems Of Complete Graph

An H-decomposition of the graph G is a partition of E(G) such that each element of the partition induces a subgraph isomorphic to H. In the case where H is an m-cycle, such a decomposition is referred to as an m-cycle system of G. An m-cycle system will be formally described as an ordered pair (V(G), B) where V(G) is the vertex set and B is the set of m-cycles.In this paper, two problems are considered:the first one is maximum hexagon packing of Kn-F, where F is a spanning forest; the second one is the spectrum for generalized almost resolvable26-cycle of complete graph Kn.This paper consists of the following three parts:In the first chapter, a short survey of an H-decomposition of the graph G is given. Some notations and symbols are also given.In the second chapter, it considers the first problem, that is maximum hexagon pack-ing of Kn-F, where F is a spanning forest. First of all, hexagon packing of some small order of graphs is given. Then by induction, the sufficient and necessary condition for maximum hexagon packing of Kn-F is proved, where F is an odd spanning forest.In the third chapter, it considers the second problem, that is the spectrum for general-ized almost resolvable26-cycle of complete graph Kn. First of all, three important examples are constructed, then by the method of differences, generalized almost resolvable26-cycle system of order n is constructed, which proves the sufficient and necessary condition for generalized almost resolvable26-cycle system of order n is n≡13(mod52)(n≥65).