Verstraëte’s almost-perfect subdivision-packing conjecture

Establish that for every graph F and every η > 0, there exists an integer d₀ = d₀(F, η) such that every d-regular graph G of order n with d ≥ d₀ contains a vertex-disjoint packing of subdivisions of F covering all but at most ηn vertices of G.

Background

The conjecture concerns packings of not-necessarily-isomorphic subdivisions of a fixed graph F in regular host graphs. It asks for an almost-perfect packing when the regular degree is sufficiently large relative to F and the permitted uncovered proportion η.

The paper proves a substantially weaker quantitative result: for degree at least (log n)^130, every sufficiently large regular graph has an F-subdivision packing leaving at most n/(log log n)^{1/30} vertices uncovered. Thus, the general conjecture remains unresolved in the form stated.