Hypertree shrinking avoiding low degree vertices
Karol\'ina Hylasov\'a, Tom\'a\v{s} Kaiser

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 hypertree can be shrunk to a tree where the degree of each vertex is at least times its degree in . We prove a stronger and a more general bound, replacing the constant with when the rank is . 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.
Peer Reviews
No public reviews on file for this paper yet. If you reviewed it on a platform where reviews are public (OpenReview, ICLR, NeurIPS, ICML), you can paste yours below so the community can read it here.
Videos
No videos yet. Explain this paper in a talk, walkthrough, or lecture? Add one.
Taxonomy
TopicsAdvanced Materials and Mechanics
