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We consider partitions of the edge set of a graph G into copies of a fixedgraph H and single edges. Let \phi_H(n) denote the minimum number p such thatany n-vertex G admits such a partition with at most p parts. We show that\phi_H(n)=ex(n,K_r)+\Theta(biex(n,H)) for \chi(H)>2, where biex(n,H) is theextremal number of the decomposition family of H. Sincebiex(n,H)=O(n^{2-\gamma}) for some \gamma>0 this improves on the bound\phi_H(n)=ex(n,H)+o(n^2) by Pikhurko and Sousa [J. Combin. Theory Ser. B 97(2007), 1041-1055]. In addition it extends a result of \"Ozkahya and Person [J.Combin. Theory Ser. B, to appear].

An improved error term for minimum H-decompositions of graphs