On Perfect Packings in Dense Graphs

We say that a graph G has a perfect H-packing if there exists a set of vertex-disjoint copies of H which cover all the vertices in G. We consider various problems concerning perfect Hpackings: Given n, r,D ∈ N, we characterise the edge density threshold that ensures a perfect Kr-packing in any graph G on n vertices and with minimum degree δ(G) ≥ D. We also give two conjectures concerning degree sequence conditions which force a graph to contain a perfect H-packing. Other related embedding problems are also considered. Indeed, we give a degree sequence condition which forces a graph to contain a copy of Kr, thereby strengthening the minimum degree version of Turán’s theorem. We also characterise the edge density threshold that ensures a graph G contains k vertex-disjoint cycles.