Packing degenerate graphs

TL;DR

The paper proves that a large family of sparse, degenerate graphs with bounded degrees can be packed into a complete graph, using a random greedy algorithm, correcting a previous error.

Contribution

It introduces a new packing result for degenerate graphs with bounded degrees, analyzed through a natural random greedy algorithm, improving understanding of graph packing.

Findings

Large families of degenerate graphs can be packed into complete graphs.

The packing is achieved via a natural random greedy algorithm.

A correction to a previously published proof is provided.

Abstract

Given $D$ and $\gamma>0$, whenever $c>0$ is sufficiently small and $n$ sufficiently large, if $\mathcal{G}$ is a family of $D$-degenerate graphs of individual orders at most $n$, maximum degrees at most $\tfrac{cn}{\log n}$, and total number of edges at most $(1-\gamma)\binom{n}{2}$, then $\mathcal{G}$ packs into the complete graph $K_{n}$. Our proof proceeds by analysing a natural random greedy packing algorithm. This version of the manuscript corrects a small error that appeared in the published version [Adv Math, 354 (2019), 106739].