Summary: 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 (H) and N(H)
such that if G is a graph with at least N(H) vertices and has minimum degree at least (1 -
(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.
All graphs considered here are finite, undirected and simple, unless otherwise noted. For the
standard graph-theoretic terminology the reader is referred to . Let H be a graph without