Packings and coverings of the complete graph with trees

In this thesis, we define the spectrum problem for packings (coverings) of G to be the problem of finding all graphs H such that a maximum G-packing (minimum G- covering) of the complete graph with the leave (excess) graph H exists. The set of achievable leave (excess) graphs in G-packings (G-coverings) of the complete graph is called the spectrum of leave (excess) graphs for G. Then, we consider this problem for trees with up to five edges. We will prove that for any tree T with up to five edges, if the leave graph in a maximum T-packing of the complete graph Kn has i edges, then the spectrum of leave graphs for T is the set of all simple graphs with i edges. In fact, for these T and i and H any simple graph with i edges, we will construct a maximum T-packing of Kn with the leave graph H. We will also show that for any tree T with k ≤ 5 edges, if the excess graph in a minimum T-covering of the complete graph Kn has i edges, then the spectrum of excess graphs for T is the set of all simple graphs and multigraphs with i edges, except for the case that T is a 5-star, for which the graph formed by four multiple edges is not achievable when n = 12.