Root-prescribed spanning loose-tree embedding

Determine whether every vertex of a sufficiently large \(k\)-uniform hypergraph with relative minimum \((k-2)\)-degree at least \(1/2+\gamma\) can serve as the image of the root in an embedding of every bounded-degree spanning \(k\)-loose tree.

Background

The main embedding theorem guarantees that a host kk-graph with relative minimum (k−2)(k-2)-degree above $1/2$ contains every spanning bounded-degree kk-loose tree when k≥4k\ge4. However, the construction does not ensure that the root of the embedded tree can be prescribed in advance.

The authors identify this limitation as arising from the reachability ordering in the robust subgraph framework and explicitly ask whether an embedding can be found with the root mapped to an arbitrary specified vertex.

References

However, our proof is unable to map the root vertex to an arbitrary vertex due to~\ref{robust2}. Then we may ask whether this is possible. \begin{ques} Let $1/n \ll \gamma \ll 1/\Delta, 1/k \le 1/4$. Let $G$ be a $k$-graph on~$n$ vertices with~$\overline{\delta}_{k-2}(G)\ge 1/2+\gamma$ and~$v\in V(G)$. Let $T$ be a rooted $n$-vertex $k$-loose tree at~$r$ with~$\Delta_1(T)\le \Delta$. Is there an embedding~$\psi$ from~$T$ to~$G$ such that $\psi(r)=v$? \end{ques}

— Embedding loose trees in $k$-uniform hypergraphs  (2502.04783 - Chen et al., 7 Feb 2025) in Concluding Remark, Section 10