Hypertree shrinking avoiding low degree vertices

TL;DR

This paper improves bounds on shrinking hypertrees to trees with controlled degrees, using hypergraph orientation techniques instead of entropy compression.

Contribution

It provides a stronger bound for hypertree shrinking, replacing the constant with a function of the hypertree's rank, and introduces new methods involving hypergraph orientations.

Findings

Bound of 1/2k for hypertree shrinking degree control

Use of hypergraph orientation lemma for the first time in this context

Characterization of rainbow spanning trees in edge-colored graphs

Abstract

The shrinking operation converts a hypergraph into a graph by choosing, from each hyperedge, two endvertices of a corresponding graph edge. A hypertree is a hypergraph which can be shrunk to a tree on the same vertex set. Klimo\v{s}ov\'{a} and Thomass\'{e} [J. Combin. Theory Ser. B 156 (2022), 250--293] proved (as a tool to obtain their main result on edge-decompositions of graphs into paths of equal length) that any rank $3$ hypertree $T$ can be shrunk to a tree where the degree of each vertex is at least $1/100$ times its degree in $T$. We prove a stronger and a more general bound, replacing the constant $1/100$ with $1/2k$ when the rank is $k$. In place of entropy compression (used by Klimo\v{s}ov\'{a} and Thomass\'{e}), we use a hypergraph orientation lemma combined with a characterisation of edge-coloured graphs admitting rainbow spanning trees.