Packing spanning graphs from separable families

TL;DR

This paper proves that large collections of graphs from a separable family with bounded maximum degree can be packed into a complete graph, improving previous results and implying approximate solutions to classical graph packing problems.

Contribution

It establishes a general packing theorem for separable graph families with bounded degree, extending known results and providing new approximate solutions to longstanding conjectures.

Findings

Graphs from separable families with bounded degree pack into complete graphs under certain conditions.

Improves bounds for packing trees and general separable graphs.

Implicates approximate solutions to the Oberwolfach problem and Tree Packing Conjecture.

Abstract

Let $\mathcal G$ be a separable family of graphs. Then for all positive constants $\epsilon$ and $\Delta$ and for every sufficiently large integer $n$, every sequence $G_1,\dotsc,G_t\in\mathcal G$ of graphs of order $n$ and maximum degree at most $\Delta$ such that $e(G_1)+\dotsb+e(G_t) \leq (1-\epsilon)\binom{n}{2}$ packs into $K_n$. This improves results of B\"ottcher, Hladk\'y, Piguet, and Taraz when $\mathcal G$ is the class of trees and of Messuti, R\"odl, and Schacht in the case of a general separable family. The result also implies approximate versions of the Oberwolfach problem and of the Tree Packing Conjecture of Gy\'arf\'as (1976) for the case that all trees have maximum degree at most $\Delta$. The proof uses the local resilience of random graphs and a special multi-stage packing procedure.