Abstract

AbstractWe show that for anyndivisible by 3, almost all order-nSteiner triple systems admit a decomposition of almost all their triples into disjoint perfect matchings (that is, almost all Steiner triple systems are almostresolvable).

Highlights

  • One of the oldest problems in combinatorics, posed by Kirkman in 1850 [24] is the following: Fifteen young ladies in a school walk out three abreast seven days in succession: it is required to arrange them daily so that no two shall walk twice abreast.This problem was solved by Kirkman a few years earlier [23].In order to frame the above question in more generality, we need to introduce some terminology.An order Steiner triple system (STS( ) for short) is a collection of triples of [ ] := {1, . . . , } for which every pair of elements is contained in exactly one triple

  • Building on the ideas in [31], introducing the sparse regularity method, and extending a random partitioning argument from [11], in this paper we prove the following ‘asymptotic’ version of Conjecture 1.1, adding to the short list of known facts about random Steiner triple systems

  • Two other relevant results are those of Frieze and Krivelevich [13] and its variant due to Ferber, Kronenberg, and Long [11], which establish a way to find ‘many’ edge-disjoint perfect matchings/Hamiltonian cycles in randomgraphs based on some pseudorandom properties and a random partitioning argument

Read more

Summary

Introduction

One of the oldest problems in combinatorics, posed by Kirkman in 1850 [24] is the following: Fifteen young ladies in a school walk out three abreast seven days in succession: it is required to arrange them daily so that no two shall walk twice abreast. Concerning the existence and completion of block designs, Kwan [31] proved that if = 3 mod 6, almost all (meaning a (1 − (1))-fraction of) order- Steiner triple systems have a perfect matching. Building on the ideas in [31], introducing the sparse regularity method, and extending a random partitioning argument from [11], in this paper we prove the following ‘asymptotic’ version of Conjecture 1.1, adding to the short list of known facts about random Steiner triple systems. Two other relevant results are those of Frieze and Krivelevich [13] and its variant due to Ferber, Kronenberg, and Long [11], which establish a way to find ‘many’ edge-disjoint perfect matchings/Hamiltonian cycles in random (hyper)graphs based on some pseudorandom properties and a random partitioning argument

Structure of the paper
Notation
Random Steiner triple systems via the triangle removal process
A coupling lemma
Sufficient properties for packing
Sparse regularity and the KŁR conjecture for hypergraphs
Refining an existing partition
The KŁR conjecture
Almost-perfect matchings
Partitioning for packing
The 3-graphs are relatively dense
Absorbers
Upper-quasirandomness
Embedding absorbers
Concluding remarks
Full Text
Published version (Free)

Talk to us

Join us for a 30 min session where you can share your feedback and ask us any queries you have

Schedule a call