On the decomposition threshold of a given graph.

We study the F -decomposition threshold δ F for a given graph F. Here an F -decomposition of a graph G is a collection of edge-disjoint copies of F in G which together cover every edge of G. (Such an F -decomposition can only exist if G is F -divisible, i.e. if e (F) | e (G) and each vertex degree of G can be expressed as a linear combination of the vertex degrees of F.) The F -decomposition threshold δ F is the smallest value ensuring that an F -divisible graph G on n vertices with δ (G) ≥ (δ F + o (1)) n has an F -decomposition. Our main results imply the following for a given graph F , where δ F ⁎ is the fractional version of δ F and χ : = χ (F) : (i) δ F ≤ max ⁡ { δ F ⁎ , 1 − 1 / (χ + 1) } ; (ii) if χ ≥ 5 , then δ F ∈ { δ F ⁎ , 1 − 1 / χ , 1 − 1 / (χ + 1) } ; (iii) we determine δ F if F is bipartite. In particular, (i) implies that δ K r = δ K r ⁎. Our proof involves further developments of the recent 'iterative' absorbing approach. [ABSTRACT FROM AUTHOR]