A fundamental theorem of Wilson states that, for every graph F, every
sufficiently large F-divisible clique has an F-decomposition. Here a graph
G is F-divisible if e(F) divides e(G) and the greatest common divisor
of the degrees of F divides the greatest common divisor of the degrees of
G, and G has an F-decomposition if the edges of G can be covered by
edge-disjoint copies of F. We extend this result to graphs G which are
allowed to be far from complete. In particular, together with a result of
Dross, our results imply that every sufficiently large K3-divisible graph of
minimum degree at least 9n/10+o(n) has a K3-decomposition. This
significantly improves previous results towards the long-standing conjecture of
Nash-Williams that every sufficiently large K3-divisible graph with minimum
degree at least 3n/4 has a K3-decomposition. We also obtain the
asymptotically correct minimum degree thresholds of 2n/3+o(n) for the
existence of a C4-decomposition, and of n/2+o(n) for the existence of a
C2ℓ-decomposition, where ℓ≥3. Our main contribution is a
general `iterative absorption' method which turns an approximate or fractional
decomposition into an exact one. In particular, our results imply that in order
to prove an asymptotic version of Nash-Williams' conjecture, it suffices to
show that every K3-divisible graph with minimum degree at least 3n/4+o(n)
has an approximate K3-decomposition,Comment: 41 pages. This version includes some minor corrections, updates and
improvement