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 ϕ_H(n) denote the minimum number p such that any n-vertex G admits such a partition with at most p parts. We show that ϕ_H(n)=ex(n,K_r)+Θ(biex(n,H)) for χ(H)>2, where biex(n,H) is the extremal number of the decomposition family of H. Since biex(n,H)=O(n^{2-γ}) for some γ>0 this improves on the bound ϕ_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 Özkahya and Person [J. Combin. Theory Ser. B, to appear].

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