Degree conditions for spanning expansion hypertrees

TL;DR

This paper establishes near-optimal degree conditions for embedding spanning $k$-expansions of bounded-degree trees in hypergraphs, revealing unexpected parity obstructions and refining previous conjectures.

Contribution

It determines asymptotically optimal degree thresholds for spanning $k$-expansions, disproving a prior conjecture and highlighting the impact of vertex degree parity.

Findings

Parity obstructions increase degree requirements for odd-degree trees.

Trees with even-degree vertices require smaller codegree conditions.

Results refute the conjecture by Pehova and Petrova.

Abstract

The $k$-expansion of a graph $G$ is the $k$-uniform hypergraph obtained from $G$ by adding $k-2$ new vertices to every edge. We determine, for all $k > d \geq 1$, asymptotically optimal $d$-degree conditions that ensure the existence of all spanning $k$-expansions of bounded-degree trees, in terms of the corresponding conditions for loose Hamilton cycles. This refutes a conjecture by Pehova and Petrova, who conjectured that a lower threshold should have sufficed. The reason why the answer is off from the conjectured value is an unexpected `parity obstruction': all spanning $k$-expansions of trees with only odd degree vertices require larger degree conditions to embed. We also show that if the tree has at least one even-degree vertex, the codegree conditions for embedding its $k$-expansion become substantially smaller.