Enumeration and randomized constructions of hypertrees

Nati Linial, Yuval Peled

Research output: Contribution to journalArticlepeer-review

Abstract

Over 30 years ago, Kalai proved a beautiful d-dimensional analog of Cayley's formula for the number of n-vertex trees. He enumerated d-dimensional hypertrees weighted by the squared size of their (d − 1)-dimensional homology group. This, however, does not answer the more basic problem of unweighted enumeration of d-hypertrees, which is our concern here. Our main result, Theorem 1.4, significantly improves the lower bound for the number of d-hypertrees. In addition, we study a random 1-out model of d-complexes where every (d − 1)-dimensional face selects a random d-face containing it, and show that it has a negligible d-dimensional homology.

Original languageEnglish
Pages (from-to)677-695
Number of pages19
JournalRandom Structures and Algorithms
Volume55
Issue number3
DOIs
StatePublished - 2019

Keywords

  • Homology
  • Hypertrees
  • Simplicial Complexes
  • The Probabilistic Method

All Science Journal Classification (ASJC) codes

  • Software
  • General Mathematics
  • Computer Graphics and Computer-Aided Design
  • Applied Mathematics

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