Perfectly packing graphs with bounded degeneracy and many leaves

Abstract

We prove that one can perfectly pack degenerate graphs into complete or dense n-vertex quasirandom graphs, provided that all the degenerate graphs have maximum degree $$o\left( {{n \over {\log \,n}}} \right)$$ , and in addition Ω(n) of them have at most (1 − Ω(1))n vertices and Ω(n) leaves. This proves Ringel’s conjecture and the Gyárfás Tree Packing Conjecture for all but an exponentially small fraction of trees (or sequences of trees, respectively).