Packing and covering dense graphs

Let d be a positive integer. A graph G is called d-divisible if d divides the degree of each vertex of G, G is called nowhere d-divisible if no degree of a vertex of G is divisible by d, For a graph H, gcd(H) denotes the greatest common divisor of the degrees of the vertices of H. The H-packing number of G is the maximum number of pairwise edge disjoint copies of H in G. The H-covering number of G is the minimum number of copies of H in G whose union covers all edges of G, Our main result is the following: For every fixed graph H with gcd(H) = d, there exist positive constants epsilon(H) and N(H) such that if G is a graph with at least N(H) vertices and has minimum degree at least (1 - epsilon(H)) \G\, then the H-packing number of G and the H-covering number of G can be computed in polynomial time. Furthermore, if G is either d-divisible or nowhere d-divisible, then there is a closed formula for the H-packing number of G, and the H-covering number of G, Further extensions and solutions to related problems are also given, (C) 1998 John Wiley & Sons, Inc. J Combin Designs 6: 451-472, 1998.