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 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 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 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 Near Perfect Matchings ink-Uniform Hypergraphs

2014 ◽  
Vol 24 (5) ◽  
pp. 723-732 ◽  
Author(s):  
JIE HAN
Keyword(s):  
Perfect Matchings ◽  
Large Integer ◽  
Uniform Hypergraph ◽  
Uniform Hypergraphs

LetHbe ak-uniform hypergraph onnvertices wherenis a sufficiently large integer not divisible byk. We prove that if the minimum (k− 1)-degree ofHis at least ⌊n/k⌋, thenHcontains a matching with ⌊n/k⌋ edges. This confirms a conjecture of Rödl, Ruciński and Szemerédi [13], who proved that minimum (k− 1)-degreen/k+O(logn) suffices. More generally, we show thatHcontains a matching of sizedif its minimum codegree isd<n/k, which is also best possible.

scholarly journals On vertex independence number of uniform hypergraphs

2014 ◽  
Vol 6 (1) ◽  
pp. 132-158 ◽  
Author(s):  
Tariq A. Chishti ◽  
Guofei Zhou ◽  
Shariefuddin Pirzada ◽  
Antal Iványi
Keyword(s):  
Independence Number ◽  
Average Degree ◽  
Uniform Hypergraph ◽  
Uniform Hypergraphs ◽  
Isolated Vertex

Abstract Let H be an r-uniform hypergraph with r ≥ 2 and let α(H) be its vertex independence number. In the paper bounds of α(H) are given for different uniform hypergraphs: if H has no isolated vertex, then in terms of the degrees, and for triangle-free linear H in terms of the order and average degree.

On Perfect Matchings in Uniform Hypergraphs with Large Minimum Vertex Degree

2009 ◽  
Vol 23 (2) ◽  
pp. 732-748 ◽  
Author(s):  
Hip Hàn ◽  
Yury Person ◽  
Mathias Schacht
Keyword(s):  
Perfect Matchings ◽  
Vertex Degree ◽  
Uniform Hypergraphs

scholarly journals Perfect Matchings in Random r-regular, s-uniform Hypergraphs

1996 ◽  
Vol 5 (1) ◽  
pp. 1-14 ◽  
Author(s):  
Colin Cooper ◽  
Alan Frieze ◽  
Michael Molloy ◽  
Bruce Reed
Keyword(s):  
High Probability ◽  
Perfect Matching ◽  
Perfect Matchings ◽  
Expected Number ◽  
Uniform Hypergraphs ◽  
The Difference

We show that r-regular, s-uniform hypergraphs contain a perfect matching with high probability (whp), provided The Proof is based on the application of a technique of Robinson and Wormald [7, 8]. The space of hypergraphs is partitioned into subsets according to the number of small cycles in the hypergraph. The difference in the expected number of perfect matchings between these subsets explains most of the variance of the number of perfect matchings in the space of hypergraphs, and is sufficient to prove existence (whp), using the Chebychev Inequality.

scholarly journals Co-degrees Resilience for Perfect Matchings in Random Hypergraphs

10.37236/8167 ◽  
2020 ◽  
Vol 27 (1) ◽  
Author(s):  
Asaf Ferber ◽  
Lior Hirschfeld
Keyword(s):  
High Probability ◽  
Perfect Matching ◽  
Perfect Matchings ◽  
Uniform Hypergraph ◽  
Hypergraph Model ◽  
Large N ◽  
Random Hypergraphs

In this paper we prove an optimal co-degrees resilience property for the binomial $k$-uniform hypergraph model $H_{n,p}^k$ with respect to perfect matchings. That is, for a sufficiently large $n$ which is divisible by $k$, and $p\geq C_k\log {n}/n$, we prove that with high probability every subgraph $H\subseteq H^k_{n,p}$ with minimum co-degree (meaning, the number of supersets every set of size $k-1$ is contained in) at least $(1/2+o(1))np$ contains a perfect matching.

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