An improved error term for minimum H-decompositions of graphs

We consider partitions of the edge set of a graph G into copies of a fixed graph H   and single edges. Let ϕH(n)ϕH(n) denote the minimum number p such that any n-vertex G admits such a partition with at most p   parts. We show that ϕH(n)=ex(n,Kr)+Θ(biex(n,H))ϕH(n)=ex(n,Kr)+Θ(biex(n,H)) for χ(H)=r≥3χ(H)=r≥3, where biex(n,H)biex(n,H) is the extremal number of the decomposition family of H  . Since biex(n,H)=O(n2−γ)biex(n,H)=O(n2−γ) for some γ>0γ>0 this improves on the bound ϕH(n)=ex(n,H)+o(n2)ϕH(n)=ex(n,H)+o(n2) by Pikhurko and Sousa (2007) [6]. In addition, it extends a result of Özkahya and Person (2012) [5].