scholarly journals Matchings and Hamilton cycles in hypergraphs

2005 ◽  
Vol DMTCS Proceedings vol. AE,... (Proceedings) ◽  
Author(s):  
Daniela Kühn ◽  
Deryk Osthus
Keyword(s):  
Bipartite Graph ◽  
Perfect Matching ◽  
Minimum Degree ◽  
Hamilton Cycle ◽  
Uniform Hypergraph ◽  
Hamilton Cycles ◽  
Uniform Hypergraphs ◽  
International Audience

International audience It is well known that every bipartite graph with vertex classes of size $n$ whose minimum degree is at least $n/2$ contains a perfect matching. We prove an analogue of this result for uniform hypergraphs. We also provide an analogue of Dirac's theorem on Hamilton cycles for $3$-uniform hypergraphs: We say that a $3$-uniform hypergraph has a Hamilton cycle if there is a cyclic ordering of its vertices such that every pair of consecutive vertices lies in a hyperedge which consists of three consecutive vertices. We prove that for every $\varepsilon > 0$ there is an $n_0$ such that every $3$-uniform hypergraph of order $n \geq n_0$ whose minimum degree is at least $n/4+ \varepsilon n$ contains a Hamilton cycle. Our bounds on the minimum degree are essentially best possible.

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 Vertex Degree Sums for Perfect Matchings in 3-Uniform Hypergraphs

10.37236/7658 ◽  
2018 ◽  
Vol 25 (3) ◽  
Author(s):  
Yi Zhang ◽  
Yi Zhao ◽  
Mei Lu
Keyword(s):  
Perfect Matching ◽  
Minimum Degree ◽  
Perfect Matchings ◽  
Vertex Degree ◽  
Uniform Hypergraph ◽  
Degree Sum ◽  
Uniform Hypergraphs ◽  
Isolated Vertex ◽  
The One

We determine the minimum degree sum of two adjacent vertices that ensures a perfect matching in a 3-uniform hypergraph without an isolated vertex. Suppose that $H$ is a 3-uniform hypergraph whose order $n$ is sufficiently large and divisible by $3$. If $H$ contains no isolated vertex and $\deg(u)+\deg(v) > \frac{2}{3}n^2-\frac{8}{3}n+2$ for any two vertices $u$ and $v$ that are contained in some edge of $H$, then $H$ contains a perfect matching. This bound is tight and the (unique) extremal hyergraph is a different space barrier from the one for the corresponding Dirac problem.

scholarly journals The complexity of 2-coloring and strong coloring in uniform hypergraphs of high minimum degree

2013 ◽  
Vol Vol. 15 no. 2 (Discrete Algorithms) ◽  
Author(s):  
Edyta Szymańska
Keyword(s):  
Time Complexity ◽  
Minimum Degree ◽  
Threshold Value ◽  
Vertex Degree ◽  
Uniform Hypergraph ◽  
Fano Plane ◽  
Uniform Hypergraphs ◽  
Discrete Algorithms ◽  
International Audience ◽  
Np Complete

Discrete Algorithms International audience In this paper we consider the problem of deciding whether a given r-uniform hypergraph H with minimum vertex degree at least c\binom|V(H)|-1r-1, or minimum degree of a pair of vertices at least c\binom|V(H)|-2r-2, has a vertex 2-coloring. Motivated by an old result of Edwards for graphs, we obtain first optimal dichotomy results for 2-colorings of r-uniform hypergraphs. For each problem, for every r≥q 3 we determine a threshold value depending on r such that the problem is NP-complete for c below the threshold, while for c strictly above the threshold it is polynomial. We provide an algorithm constructing the coloring with time complexity O(n^\lfloor 4/ε\rfloor+2\log n) with some ε>0. This algorithm becomes more efficient in the case of r=3,4,5 due to known Turán numbers of the triangle and the Fano plane. In addition, we determine the computational complexity of strong k-coloring of 3-uniform hypergraphs H with minimum vertex degree at least c\binom|V(H)|-12, for some c, leaving a gap for k≥q 5 which vanishes as k→ ∞.

scholarly journals Rainbow Hamilton Cycles in Uniform Hypergraphs

10.37236/2055 ◽  
2012 ◽  
Vol 19 (1) ◽  
Author(s):  
Andrzej Dudek ◽  
Alan Frieze ◽  
Andrzej Ruciński
Keyword(s):  
Maximum Degree ◽  
Hamilton Cycle ◽  
Uniform Hypergraph ◽  
Hamilton Cycles ◽  
Lovász Local Lemma ◽  
Uniform Hypergraphs ◽  
Local Lemma ◽  
Large N ◽  
Different Color ◽  
Open Question

Let $K_n^{(k)}$ be the complete $k$-uniform hypergraph, $k\ge3$, and let $\ell$ be an integer such that $1\le \ell\le k-1$ and $k-\ell$ divides $n$. An $\ell$-overlapping Hamilton cycle in $K_n^{(k)}$ is a spanning subhypergraph $C$ of  $K_n^{(k)}$  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.We show that, for some constant $c=c(k,\ell)$ and sufficiently large $n$, for every coloring (partition) of the edges of $K_n^{(k)}$ which uses arbitrarily many colors but no color appears more than $cn^{k-\ell}$ times, there exists a rainbow $\ell$-overlapping Hamilton cycle $C$, that is every edge of $C$ receives a different color. We also prove that, for some constant $c'=c'(k,\ell)$ and sufficiently large $n$, for every coloring of the edges of $K_n^{(k)}$ in which the maximum degree of the subhypergraph induced by any single color is bounded by $c'n^{k-\ell}$,  there exists a properly colored $\ell$-overlapping Hamilton cycle $C$, that is every two adjacent edges receive different colors. For $\ell=1$, both results are (trivially) best possible up to the constants. It is an open question if our results are also optimal for $2\le\ell\le k-1$.The proofs  rely on a version of the Lovász Local Lemma and incorporate some ideas from Albert, Frieze, and Reed.

scholarly journals Loose Hamilton Cycles in Random Uniform Hypergraphs

10.37236/535 ◽  
2011 ◽  
Vol 18 (1) ◽  
Author(s):  
Andrzej Dudek ◽  
Alan Frieze
Keyword(s):  
Hamilton Cycle ◽  
Uniform Hypergraph ◽  
Hamilton Cycles ◽  
Single Vertex ◽  
Uniform Hypergraphs

In the random $k$-uniform hypergraph $H_{n,p;k}$ 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 adjacent edges intersects in a single vertex. We prove that if $p n^{k-1}/\log n$ tends to infinity with $n$ then $$\lim_{\substack{n\to \infty\\ 2(k-1) |n}}\Pr(H_{n,p;k}\text{ contains a loose Hamilton cycle})=1.$$ This is asymptotically best possible.

scholarly journals Cycles and Matchings in Randomly Perturbed Digraphs and Hypergraphs

2016 ◽  
Vol 25 (6) ◽  
pp. 909-927 ◽  
Author(s):  
MICHAEL KRIVELEVICH ◽  
MATTHEW KWAN ◽  
BENNY SUDAKOV
Keyword(s):  
Perfect Matching ◽  
Minimum Degree ◽  
Random Perturbation ◽  
Hamilton Cycle ◽  
Regularity Lemma ◽  
Hamilton Cycles ◽  
Interesting Application ◽  
Discrete Structures ◽  
Szemeredi's Regularity Lemma ◽  
Strong Expansion

We give several results showing that different discrete structures typically gain certain spanning substructures (in particular, Hamilton cycles) after a modest random perturbation. First, we prove that adding linearly many random edges to a densek-uniform hypergraph ensures the (asymptotically almost sure) existence of a perfect matching or a loose Hamilton cycle. The proof involves an interesting application of Szemerédi's Regularity Lemma, which might be independently useful. We next prove that digraphs with certain strong expansion properties are pancyclic, and use this to show that adding a linear number of random edges typically makes a dense digraph pancyclic. Finally, we prove that perturbing a certain (minimum-degree-dependent) number of random edges in a tournament typically ensures the existence of multiple edge-disjoint Hamilton cycles. All our results are tight.

scholarly journals Counting Hamilton cycles in Dirac hypergraphs

2020 ◽  
pp. 1-23
Author(s):  
Stefan Glock ◽  
Stephen Gould ◽  
Felix Joos ◽  
Daniela Kühn ◽  
Deryk Osthus
Keyword(s):  
Hamilton Cycle ◽  
Uniform Hypergraph ◽  
Similar Estimate ◽  
Hamilton Cycles

Abstract A tight Hamilton cycle in a k-uniform hypergraph (k-graph) G is a cyclic ordering of the vertices of G such that every set of k consecutive vertices in the ordering forms an edge. Rödl, Ruciński and Szemerédi proved that for $k\ge 3$ , every k-graph on n vertices with minimum codegree at least $n/2+o(n)$ contains a tight Hamilton cycle. We show that the number of tight Hamilton cycles in such k-graphs is ${\exp(n\ln n-\Theta(n))}$ . As a corollary, we obtain a similar estimate on the number of Hamilton ${\ell}$ -cycles in such k-graphs for all ${\ell\in\{0,\ldots,k-1\}}$ , which makes progress on a question of Ferber, Krivelevich and Sudakov.

scholarly journals An approximate version of Jackson’s conjecture

2020 ◽  
Vol 29 (6) ◽  
pp. 886-899
Author(s):  
Anita Liebenau ◽  
Yanitsa Pehova
Keyword(s):  
Bipartite Graph ◽  
Small Proportion ◽  
Complete Bipartite Graph ◽  
Hamilton Cycle ◽  
Hamilton Cycles ◽  
Edge Disjoint ◽  
Bipartite Digraph ◽  
Bipartite Tournament ◽  
Approximate Version

AbstractA diregular bipartite tournament is a balanced complete bipartite graph whose edges are oriented so that every vertex has the same in- and out-degree. In 1981 Jackson showed that a diregular bipartite tournament contains a Hamilton cycle, and conjectured that in fact its edge set can be partitioned into Hamilton cycles. We prove an approximate version of this conjecture: for every ε > 0 there exists n0 such that every diregular bipartite tournament on 2n ≥ n0 vertices contains a collection of (1/2–ε)n cycles of length at least (2–ε)n. Increasing the degree by a small proportion allows us to prove the existence of many Hamilton cycles: for every c > 1/2 and ε > 0 there exists n0 such that every cn-regular bipartite digraph on 2n ≥ n0 vertices contains (1−ε)cn edge-disjoint Hamilton cycles.

scholarly journals Edge Disjoint Hamilton Cycles in Knödel Graphs

2016 ◽  
Vol Vol. 17 no. 3 (Graph Theory) ◽  
Author(s):  
Palanivel Subramania Nadar Paulraja ◽  
S Sampath Kumar
Keyword(s):  
Maximum Degree ◽  
Hamilton Cycle ◽  
Hamilton Cycles ◽  
Cycle Decomposition ◽  
Edge Disjoint ◽  
International Audience ◽  
Appl Math

International audience The vertices of the Knödel graph $W_{\Delta, n}$ on $n \geq 2$ vertices, $n$ even, and of maximum degree $\Delta, 1 \leq \Delta \leq \lfloor log_2(n) \rfloor$, are the pairs $(i,j)$ with $i=1,2$ and $0 \leq j \leq \frac{n}{2} -1$. For $0 \leq j \leq \frac{n}{2} -1$, there is an edge between vertex $(1,j)$ and every vertex $(2,j + 2^k - 1 (mod \frac{n}{2}))$, for $k=0,1,2, \ldots , \Delta -1$. Existence of a Hamilton cycle decomposition of $W_{k, 2k}, k \geq 6$ is not yet known, see Discrete Appl. Math. 137 (2004) 173-195. In this paper, it is shown that the $k$-regular Knödel graph $W_{k,2k}, k \geq 6$ has $ \lfloor \frac{k}{2} \rfloor - 1$ edge disjoint Hamilton cycles.

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