scholarly journals Minimum vertex degree conditions for loose Hamilton cycles in 3-uniform hypergraphs

2011 ◽  
Vol 38 ◽  
pp. 207-212
Author(s):  
Enno Buß ◽  
Hiệp Hàn ◽  
Mathias Schacht
Keyword(s):  
Vertex Degree ◽  
Hamilton Cycles ◽  
Uniform Hypergraphs ◽  
Degree Conditions

scholarly journals Resilient Degree Sequences with respect to Hamilton Cycles and Matchings in Random Graphs

10.37236/8279 ◽  
2019 ◽  
Vol 26 (4) ◽  
Author(s):  
Padraig Condon ◽  
Alberto Espuny Díaz ◽  
Daniela Kühn ◽  
Deryk Osthus ◽  
Jaehoon Kim
Keyword(s):  
Random Graphs ◽  
Perfect Matching ◽  
Degree Sequence ◽  
Hamilton Cycle ◽  
Vertex Degree ◽  
Degree Sequences ◽  
Hamilton Cycles ◽  
Degree Condition ◽  
Degree Conditions

Pósa's theorem states that any graph $G$ whose degree sequence $d_1 \le \cdots \le d_n$ satisfies $d_i \ge i+1$ for all $i < n/2$ has a Hamilton cycle. This degree condition is best possible. We show that a similar result holds for suitable subgraphs $G$ of random graphs, i.e.~we prove a `resilience version' of Pósa's theorem: if $pn \ge C \log n$ and the $i$-th vertex degree (ordered increasingly) of $G \subseteq G_{n,p}$ is at least $(i+o(n))p$ for all $i<n/2$, then $G$ has a Hamilton cycle. This is essentially best possible and strengthens a resilience version of Dirac's theorem obtained by Lee and Sudakov. Chvátal's theorem generalises Pósa's theorem and characterises all degree sequences which ensure the existence of a Hamilton cycle. We show that a natural guess for a resilience version of Chvátal's theorem fails to be true. We formulate a conjecture which would repair this guess, and show that the corresponding degree conditions ensure the existence of a perfect matching in any subgraph of $G_{n,p}$ which satisfies these conditions. This provides an asymptotic characterisation of all degree sequences which resiliently guarantee the existence of a perfect matching.

Localized Codegree Conditions for Tight Hamilton Cycles in 3-Uniform Hypergraphs

2022 ◽  
Vol 36 (1) ◽  
pp. 147-169
Author(s):  
Pedro Araújo ◽  
Simón Piga ◽  
Mathias Schacht
Keyword(s):  
Hamilton Cycles ◽  
Uniform Hypergraphs

scholarly journals Triangle-degrees in graphs and tetrahedron coverings in 3-graphs

2020 ◽  
pp. 1-25
Author(s):  
Victor Falgas-Ravry ◽  
Klas Markström ◽  
Yi Zhao
Keyword(s):  
Upper Bound ◽  
Vertex Degree ◽  
Covering Problem ◽  
Uniform Hypergraphs ◽  
Vertex Graph ◽  
Optimal Bounds ◽  
Special Instance ◽  
Tripartite Graphs

Abstract We investigate a covering problem in 3-uniform hypergraphs (3-graphs): Given a 3-graph F, what is c1(n, F), the least integer d such that if G is an n-vertex 3-graph with minimum vertex-degree $\delta_1(G)>d$ then every vertex of G is contained in a copy of F in G? We asymptotically determine c1(n, F) when F is the generalized triangle $K_4^{(3)-}$ , and we give close to optimal bounds in the case where F is the tetrahedron $K_4^{(3)}$ (the complete 3-graph on 4 vertices). This latter problem turns out to be a special instance of the following problem for graphs: Given an n-vertex graph G with $m> n^2/4$ edges, what is the largest t such that some vertex in G must be contained in t triangles? We give upper bound constructions for this problem that we conjecture are asymptotically tight. We prove our conjecture for tripartite graphs, and use flag algebra computations to give some evidence of its truth in the general case.

scholarly journals Packing tight Hamilton cycles in 3-uniform hypergraphs

2011 ◽  
Vol 40 (3) ◽  
pp. 269-300 ◽  
Author(s):  
Alan Frieze ◽  
Michael Krivelevich ◽  
Po-Shen Loh
Keyword(s):  
Hamilton Cycles ◽  
Uniform Hypergraphs

scholarly journals Vertex Degree Sums for Matchings in 3-Uniform Hypergraphs

10.37236/8627 ◽  
2019 ◽  
Vol 26 (4) ◽  
Author(s):  
Yi Zhang ◽  
Yi Zhao ◽  
Mei Lu
Keyword(s):  
Vertex Degree ◽  
Uniform Hypergraph ◽  
Degree Sum ◽  
Uniform Hypergraphs ◽  
Positive Integers ◽  
Degree Sum Condition

Let $n, s$ be positive integers such that $n$ is sufficiently large and $s\le n/3$. Suppose $H$ is a 3-uniform hypergraph of order $n$ without isolated vertices. If $\deg(u)+\deg(v) > 2(s-1)(n-1)$ for any two vertices $u$ and $v$ that are contained in some edge of $H$, then $H$ contains a matching of size $s$. This degree sum condition is best possible and confirms a conjecture of the authors [Electron. J. Combin. 25 (3), 2018], who proved the case when $s= n/3$.

scholarly journals Loose Hamilton Cycles in Random 3-Uniform Hypergraphs

10.37236/477 ◽  
2010 ◽  
Vol 17 (1) ◽  
Author(s):  
Alan Frieze
Keyword(s):  
Absolute Constant ◽  
Hamilton Cycle ◽  
Hamilton Cycles ◽  
Uniform Hypergraphs ◽  
Random Hypergraph

In the random hypergraph $H=H_{n,p;3}$ each possible triple appears independently with probability $p$. A loose Hamilton cycle can be described as a sequence of edges $\{x_i,y_i,x_{i+1}\}$ for $i=1,2,\ldots,n/2$ where $x_1,x_2,\ldots,x_{n/2},y_1,y_2,\ldots,y_{n/2}$ are all distinct. We prove that there exists an absolute constant $K>0$ such that if $p\geq {K\log n\over n^2}$ then $$\lim_{\textstyle{n\to \infty\atop 4|n}}\Pr(H_{n,p;3}\ contains\ a\ loose\ Hamilton\ cycle)=1.$$

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.

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