Packing tree factors in random and pseudo-random graphs

TL;DR

This paper proves that in random and pseudo-random graphs, large collections of edge-disjoint tree factors can be packed to cover almost all edges, under certain probabilistic and degree conditions.

Contribution

It establishes new packing thresholds for tree factors in random and pseudo-random graphs, extending previous results to broader settings.

Findings

High probability packing of T-factors in G_{n,p} for p > (C log n)/n

Almost complete edge coverage with T-factors in pseudo-random graphs

Conditions under which the packing is optimal or near-optimal

Abstract

For 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 \epsilon > 0, the random graph G_{n,p}, with n a multiple of t, with high probability contains a family of edge-disjoint T-factors covering all but an \epsilon-fraction of its edges, as long as \epsilon^4 n p >> (log n)^2. Assuming stronger divisibility conditions, the edge probability can be taken down to p > (C log n)/n. A similar packing result is proved also for pseudo-random graphs, defined in terms of their degrees and co-degrees.