# Packing Tree Factors in Random and Pseudo-Random Graphs

journal contribution

posted on 01.04.2014, 00:00 by Deepak Bal, Alan Frieze, Michael Krivelevich, Po-Shen LohFor a fixed graph H with t vertices, an H-factor of a graph G with n vertices, where t divides n, is a collection of vertex disjoint (not necessarily induced) copies of H in G covering all vertices of G. We prove that for a fixed tree T on t vertices and ϵ>0, the random graph Gn,p, with n a multiple of t, with high probability contains a family of edge-disjoint T-factors covering all but an ϵ-fraction of its edges, as long as ϵ4np≫log2n. Assuming stronger divisibility conditions, the edge probability can be taken down to p>Clognn. A similar packing result is proved also for pseudo-random graphs, defined in terms of their degrees and co-degrees.