scholarly journals Tight Hamilton cycles in random hypergraphs

2013 ◽  
Vol 46 (3) ◽  
pp. 446-465 ◽  
Author(s):  
Peter Allen ◽  
Julia Böttcher ◽  
Yoshiharu Kohayakawa ◽  
Yury Person
Keyword(s):  
Hamilton Cycles ◽  
Random Hypergraphs

scholarly journals Powers of Hamilton cycles in random graphs and tight Hamilton cycles in random hypergraphs

2018 ◽  
Vol 54 (1) ◽  
pp. 187-208 ◽  
Author(s):  
Rajko Nenadov ◽  
Nemanja Škorić
Keyword(s):  
Random Graphs ◽  
Hamilton Cycles ◽  
Random Hypergraphs

scholarly journals Factors and loose Hamilton cycles in sparse pseudo-random hypergraphs

2020 ◽  
pp. 702-717 ◽  
Author(s):  
Hiệp Hàn ◽  
Jie Han ◽  
Patrick Morris
Keyword(s):  
Hamilton Cycles ◽  
Random Hypergraphs

scholarly journals Optimal Divisibility Conditions for Loose Hamilton Cycles in Random Hypergraphs

10.37236/2523 ◽  
2012 ◽  
Vol 19 (4) ◽  
Author(s):  
Andrzej Dudek ◽  
Alan Frieze ◽  
Po-Shen Loh ◽  
Shelley Speiss
Keyword(s):  
Absolute Constant ◽  
Hamilton Cycle ◽  
Uniform Hypergraph ◽  
Hamilton Cycles ◽  
Single Vertex ◽  
Random Hypergraphs

In the random $k$-uniform hypergraph $H^{(k)}_{n,p}$ of order $n$, each possible $k$-tuple appears independently with probability $p$. A loose Hamilton cycle is a cycle of order $n$ in which every pair of consecutive edges intersects in a single vertex. It was shown by Frieze that if $p\geq c(\log n)/n^2$ for some absolute constant $c>0$, then a.a.s. $H^{(3)}_{n,p}$ contains a loose Hamilton cycle, provided that $n$ is divisible by $4$. Subsequently,  Dudek and Frieze extended this result for any uniformity $k\ge 4$, proving that if $p\gg (\log n)/n^{k-1}$, then $H^{(k)}_{n,p}$ contains a loose Hamilton cycle, provided that $n$ is divisible by $2(k-1)$. In this paper, we improve the divisibility requirement and show that in the above results it is enough to assume that $n$ is a multiple of $k-1$, which is best possible.

scholarly journals Non-linear Hamilton cycles in linear quasi-random hypergraphs

2021 ◽  
pp. 74-88
Author(s):  
Jie Han ◽  
Xichao Shu ◽  
Guanghui Wang
Keyword(s):  
Hamilton Cycles ◽  
Non Linear ◽  
Random Hypergraphs

scholarly journals Factors and loose Hamilton cycles in sparse pseudo‐random hypergraphs

2021 ◽  
Author(s):  
Hiệp Hàn ◽  
Jie Han ◽  
Patrick Morris
Keyword(s):  
Hamilton Cycles ◽  
Random Hypergraphs

scholarly journals Finding Tight Hamilton Cycles in Random Hypergraphs Faster

2018 ◽  
pp. 28-36
Author(s):  
Peter Allen ◽  
Christoph Koch ◽  
Olaf Parczyk ◽  
Yury Person
Keyword(s):  
Hamilton Cycles ◽  
Random Hypergraphs

scholarly journals Finding tight Hamilton cycles in random hypergraphs faster

2020 ◽  
pp. 1-19
Author(s):  
Peter Allen ◽  
Christoph Koch ◽  
Olaf Parczyk ◽  
Yury Person
Keyword(s):  
Polynomial Time ◽  
Polynomial Time Algorithm ◽  
Time Algorithm ◽  
Hamilton Cycle ◽  
Uniform Hypergraph ◽  
Hamilton Cycles ◽  
Second Moment ◽  
Uniform Hypergraphs ◽  
Edge Probability ◽  
Random Hypergraphs

Abstract In an r-uniform hypergraph on n vertices, a tight Hamilton cycle consists of n edges such that there exists a cyclic ordering of the vertices where the edges correspond to consecutive segments of r vertices. We provide a first deterministic polynomial-time algorithm, which finds a.a.s. tight Hamilton cycles in random r-uniform hypergraphs with edge probability at least C log3n/n. Our result partially answers a question of Dudek and Frieze, who proved that tight Hamilton cycles exist already for p = ω(1/n) for r = 3 and p = (e + o(1))/n for $r \ge 4$ using a second moment argument. Moreover our algorithm is superior to previous results of Allen, Böttcher, Kohayakawa and Person, and Nenadov and Škorić, in various ways: the algorithm of Allen et al. is a randomized polynomial-time algorithm working for edge probabilities $p \ge {n^{ - 1 + \varepsilon}}$ , while the algorithm of Nenadov and Škorić is a randomized quasipolynomial-time algorithm working for edge probabilities $p \ge C\mathop {\log }\nolimits^8 n/n$ .

scholarly journals Packing hamilton cycles in random and pseudo-random hypergraphs

2012 ◽  
Vol 41 (1) ◽  
pp. 1-22 ◽  
Author(s):  
Alan Frieze ◽  
Michael Krivelevich
Keyword(s):  
Hamilton Cycles ◽  
Random Hypergraphs

scholarly journals On Rainbow Hamilton Cycles in Random Hypergraphs

10.37236/7274 ◽  
2018 ◽  
Vol 25 (2) ◽  
Author(s):  
Andrzej Dudek ◽  
Sean English ◽  
Alan Frieze
Keyword(s):  
Hamilton Cycle ◽  
Uniform Hypergraph ◽  
Hamilton Cycles ◽  
Sharp Threshold ◽  
Vertex Set ◽  
Different Color ◽  
Random Hypergraph ◽  
Random Hypergraphs

Let $H_{n,p,r}^{(k)}$ denote a randomly colored random hypergraph, constructed on the vertex set $[n]$ by taking each $k$-tuple independently with probability $p$, and then independently coloring it with a random color from the set $[r]$. Let $H$ be a $k$-uniform hypergraph of order $n$. An $\ell$-Hamilton cycle is a spanning subhypergraph $C$ of $H$ with $n/(k-\ell)$ edges and such that for some cyclic ordering of the vertices each edge of $C$ consists of $k$ consecutive vertices and every pair of adjacent edges in $C$ intersects in precisely $\ell$ vertices.In this note we study the existence of rainbow $\ell$-Hamilton cycles (that is every edge receives a different color) in $H_{n,p,r}^{(k)}$. We mainly focus on the most restrictive case when $r = n/(k-\ell)$. In particular, we show that for the so called tight Hamilton cycles ($\ell=k-1$) $p = e^2/n$ is the sharp threshold for the existence of a rainbow tight Hamilton cycle in $H_{n,p,n}^{(k)}$ for each $k\ge 4$.

scholarly journals On offset Hamilton cycles in random hypergraphs

2018 ◽  
Vol 238 ◽  
pp. 77-85 ◽  
Author(s):  
Andrzej Dudek ◽  
Laars Helenius
Keyword(s):  
Hamilton Cycles ◽  
Random Hypergraphs
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