Perfectly packing graphs with bounded degeneracy and many leaves

TL;DR

This paper proves that a wide class of degenerate graphs, including nearly all trees, can be perfectly packed into dense quasirandom graphs under certain degree and leaf conditions, confirming longstanding conjectures.

Contribution

It establishes the first near-complete proof of Ringel's conjecture and the Gyárfás Tree Packing Conjecture for almost all trees by allowing maximum degree up to o(n/log n) and many leaves.

Findings

Proves perfect packing of degenerate graphs with maximum degree o(n/log n).

Confirms Ringel's conjecture for almost all trees.

Validates Gyárfás Tree Packing Conjecture for an exponential fraction of trees.

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 $o(\frac{n}{\log n})$, and in addition $\Omega(n)$ of them have at most $(1-\Omega(1))n$ vertices and $\Omega(n)$ leaves. This proves Ringel's conjecture and the Gy\'arf\'as Tree Packing Conjecture for all but an exponentially small fraction of trees (or sequences of trees, respectively).