The asymptotic minimum-degree characterization for H-factors with acyclic-partition hcf 1

Let $H$ be a graph, let $\mathrm{AP}$ denote the family of all acyclic partitions of $H$ with $ar(H)$ parts, and let $\mathrm{hcf}(H,\mathrm{AP})$ be the associated gcd parameter. For a graph $H$, define its critical arboricity by

$$ar_{cr}(H)=\frac{(ar(H)-1)|V(H)|}{|V(H)|-\sigma(H)},$$

where $\sigma(H)$ is the minimum size of a part over all optimal acyclic partitions of $H$.

The asymptotic $H$-factor conjecture. Given $\mu>0$, $h\in\mathbb{N}$, and an $h$-vertex graph $H$ with $\mathrm{hcf}(H,\mathrm{AP})=1$, there exists $\alpha>0$ such that, for all sufficiently large $n\in h\mathbb{N}$, every $n$-vertex graph $G$ satisfying

$$\delta(G)\geq\max\left\{\left(1-\frac{1}{ar_{cr}(H)}+\mu\right)n,\left(\frac{1}{2}+\mu\right)n\right\}$$

and $\alpha(G)\leq\alpha n$ contains an $H$-factor.

The conjecture proposes the missing converse to the paper's asymptotic sharpness results for the minimum-degree condition in the $H$-factor problem. The lower bound from the space barrier establishes sharpness for graphs with critical arboricity attaining the relevant threshold, while the case $\mathrm{hcf}(H,\mathrm{AP})\neq1$ is already covered by the stated dichotomy; the general $\mathrm{hcf}(H,\mathrm{AP})=1$ case remains open.