Packing bounded-degree spanning graphs from separable families

Let G be a separable family of graphs. Then for all positive constants and ∆ and for every su ciently large integer n, every sequence G1, . . . , Gt ∈ G of graphs of order n and maximum degree at most ∆ such that e(G1)+· · ·+e(Gt) ≤ (1− ) ( n 2 ) packs into Kn. This improves results of Böttcher, Hladký, Piguet, and Taraz when G is the class of trees and of Messuti, Rödl, and Schacht in the case of a general separable family. The result also implies an approximate version of the Tree Packing Conjecture of Gyárfás (1976) for the case that all trees have maximum degree at most ∆. The proof uses well-known properties of random graphs and a special multi-stage packing procedure.