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Sharp thresholds for nonlinear Hamiltonian cycles in hyerpgraphs
Random Structures and Algorithms ( IF 0.9 ) Pub Date : 2020-02-27 , DOI: 10.1002/rsa.20919 Bhargav Narayanan 1 , Mathias Schacht 2
Random Structures and Algorithms ( IF 0.9 ) Pub Date : 2020-02-27 , DOI: 10.1002/rsa.20919 Bhargav Narayanan 1 , Mathias Schacht 2
Affiliation
For positive integers r >ℓ , an r ‐uniform hypergraph is called an ℓ ‐cycle if there exists a cyclic ordering of its vertices such that each of its edges consists of r consecutive vertices, and such that every pair of consecutive edges (in the natural ordering of the edges) intersect in precisely ℓ vertices; such cycles are said to be linear when ℓ =1, and nonlinear when ℓ >1. We determine the sharp threshold for nonlinear Hamiltonian cycles and show that for all r >ℓ >1, the threshold for the appearance of a Hamiltonian ℓ ‐cycle in the random r ‐uniform hypergraph on n vertices is sharp and given by for an explicitly specified function λ . This resolves several questions raised by Dudek and Frieze in 2011.10
中文翻译:
曲线图中非线性哈密顿循环的尖锐阈值
为正整数ř > ℓ,一个ř -uniform超图被称为ℓ如果存在其顶点的环状排序,使得其各个边缘的由-cycle ř连续的顶点,并且使得每对连续的边缘的(在边缘的自然顺序)恰好相交于ℓ个顶点;这种循环被认为是线性时ℓ = 1,且当非线性ℓ > 1。我们确定了非线性哈密顿循环的尖锐阈值,并表明对于所有r > ℓ > 1,哈密顿ian出现的阈值在n个顶点上的随机r均匀超图中的循环是尖锐的,并且由明确指定的函数λ给出。这解决了Dudek和Frieze在2011.10中提出的几个问题。
更新日期:2020-02-27
中文翻译:
曲线图中非线性哈密顿循环的尖锐阈值
为正整数ř > ℓ,一个ř -uniform超图被称为ℓ如果存在其顶点的环状排序,使得其各个边缘的由-cycle ř连续的顶点,并且使得每对连续的边缘的(在边缘的自然顺序)恰好相交于ℓ个顶点;这种循环被认为是线性时ℓ = 1,且当非线性ℓ > 1。我们确定了非线性哈密顿循环的尖锐阈值,并表明对于所有r > ℓ > 1,哈密顿ian出现的阈值在n个顶点上的随机r均匀超图中的循环是尖锐的,并且由明确指定的函数λ给出。这解决了Dudek和Frieze在2011.10中提出的几个问题。