Edge-decompositions of graphs with high minimum degree

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 which are allowed to be far from complete: our results imply that every sufficiently large F -divisible graph G on n vertices with minimum degree at least (1− 1/(16|F |(|F | − 1)) + ε)n has an F -decomposition. Moreover, every sufficiently large K3-divisible graph of minimum degree at least 0.956n has a K3-decomposition. Our result 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. For certain graphs, we can strengthen the general bound above. In particular, we obtain the asymptotically correct thresholds of 2n/3 + o(n) for C4 and n/2 + o(n) for even cycles of length at least 6. Our main contribution is a general ‘iterative absorption’ method which turns an approximate decomposition into an exact one.