Packing subdivisions into regular graphs

TL;DR

This paper proves that large regular graphs can be almost entirely covered by vertex-disjoint subdivisions of any fixed graph, confirming a long-standing conjecture and improving previous bounds on the degree requirement.

Contribution

It establishes that for any fixed graph and small error margin, large regular graphs can be nearly fully covered by subdivisions, confirming Verstra"ete's 2002 conjecture.

Findings

Verifies Verstra"ete's conjecture from 2002.

Shows coverage of at least (1-η)n vertices by subdivisions.

Reduces degree requirement from polylogarithmic to a fixed threshold.

Abstract

We show that, for any graph $F$ and $\eta>0$, there exists a $d_0=d_0(F,\eta)$ such that every $n$-vertex $d$-regular graph with $d \geq d_0$ has a collection of vertex-disjoint $F$-subdivisions covering at least $(1-\eta)n$ vertices. This verifies a conjecture of Verstra\"ete from 2002 and improves a recent result of Letzter, Methuku and Sudakov which additionally required $d$ to be at least polylogarithmic in $n$.