scholarly journals The minimum vertex degree for an almost-spanning tight cycle in a 3-uniform hypergraph

2017 ◽  
Vol 340 (6) ◽  
pp. 1172-1179 ◽  
Author(s):  
Oliver Cooley ◽  
Richard Mycroft
Keyword(s):  
Vertex Degree ◽  
Uniform Hypergraph

scholarly journals A Separator Theorem for Hypergraphs and a CSP-SAT Algorithm

2021 ◽  
Vol Volume 17, Issue 4 ◽  
Author(s):  
Michal Koucký ◽  
Vojtěch Rödl ◽  
Navid Talebanfard
Keyword(s):  
Deterministic Algorithm ◽  
Vertex Degree ◽  
Connected Components ◽  
Variable Frequency ◽  
Uniform Hypergraph ◽  
Separator Theorem

We show that for every $r \ge 2$ there exists $\epsilon_r > 0$ such that any $r$-uniform hypergraph with $m$ edges and maximum vertex degree $o(\sqrt{m})$ contains a set of at most $(\frac{1}{2} - \epsilon_r)m$ edges the removal of which breaks the hypergraph into connected components with at most $m/2$ edges. We use this to give an algorithm running in time $d^{(1 - \epsilon_r)m}$ that decides satisfiability of $m$-variable $(d, k)$-CSPs in which every variable appears in at most $r$ constraints, where $\epsilon_r$ depends only on $r$ and $k\in o(\sqrt{m})$. Furthermore our algorithm solves the corresponding #CSP-SAT and Max-CSP-SAT of these CSPs. We also show that CNF representations of unsatisfiable $(2, k)$-CSPs with variable frequency $r$ can be refuted in tree-like resolution in size $2^{(1 - \epsilon_r)m}$. Furthermore for Tseitin formulas on graphs with degree at most $k$ (which are $(2, k)$-CSPs) we give a deterministic algorithm finding such a refutation.

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 $s= n/3$.

Minimum Degrees and Codegrees of Ramsey-Minimal 3-Uniform Hypergraphs

2016 ◽  
Vol 25 (6) ◽  
pp. 850-869
Author(s):  
DENNIS CLEMENS ◽  
YURY PERSON
Keyword(s):  
Vertex Degree ◽  
Uniform Hypergraph ◽  
Minimal Graphs ◽  
Uniform Hypergraphs ◽  
Minimum Degrees ◽  
Monochromatic Copy

A uniform hypergraph H is called k-Ramsey for a hypergraph F if, no matter how one colours the edges of H with k colours, there is always a monochromatic copy of F. We say that H is k-Ramsey-minimal for F if H is k-Ramsey for F but every proper subhypergraph of H is not. Burr, Erdős and Lovasz studied various parameters of Ramsey-minimal graphs. In this paper we initiate the study of minimum degrees and codegrees of Ramsey-minimal 3-uniform hypergraphs. We show that the smallest minimum vertex degree over all k-Ramsey-minimal 3-uniform hypergraphs for Kt(3) is exponential in some polynomial in k and t. We also study the smallest possible minimum codegree over 2-Ramsey-minimal 3-uniform hypergraphs.

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 A Dirac-type Theorem for Berge Cycles in Random Hypergraphs

10.37236/8611 ◽  
2020 ◽  
Vol 27 (3) ◽  
Author(s):  
Dennis Clemens ◽  
Julia Ehrenmüller ◽  
Yury Person
Keyword(s):  
Minimum Degree ◽  
Type Theorem ◽  
Vertex Degree ◽  
Uniform Hypergraph ◽  
Degree Condition ◽  
Dirac Type ◽  
Spanning Subgraph ◽  
Random Hypergraphs

A Hamilton Berge cycle of a hypergraph on $n$ vertices is an alternating sequence $(v_1, e_1, v_2, \ldots, v_n, e_n)$ of distinct vertices $v_1, \ldots, v_n$ and distinct hyperedges $e_1, \ldots, e_n$ such that $\{v_1,v_n\}\subseteq e_n$ and $\{v_i, v_{i+1}\} \subseteq e_i$ for every $i\in [n-1]$. We prove the following Dirac-type theorem about Berge cycles in the binomial random $r$-uniform hypergraph $H^{(r)}(n,p)$: for every integer $r \geq 3$, every real $\gamma>0$ and $p \geq \frac{\ln^{17r} n}{n^{r-1}}$ asymptotically almost surely,  every spanning subgraph $H \subseteq H^{(r)}(n,p)$ with  minimum vertex degree $\delta_1(H) \geq \left(\frac{1}{2^{r-1}} + \gamma\right) p \binom{n}{r-1}$ contains a Hamilton Berge cycle. The minimum degree condition is asymptotically tight and the bound on $p$ is optimal up to some polylogarithmic factor.  

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 Wiener–Hosoya Matrix of Connected Graphs

Mathematics ◽  
2021 ◽  
Vol 9 (4) ◽  
pp. 359
Author(s):  
Hassan Ibrahim ◽  
Reza Sharafdini ◽  
Tamás Réti ◽  
Abolape Akwu
Keyword(s):  
Lower Bounds ◽  
Molecular Graph ◽  
Wiener Index ◽  
Vertex Degree ◽  
Upper And Lower Bounds ◽  
Connected Graphs ◽  
Matrix Description ◽  
Vertex Set

Let G be a connected (molecular) graph with the vertex set V(G)={v1,⋯,vn}, and let di and σi denote, respectively, the vertex degree and the transmission of vi, for 1≤i≤n. In this paper, we aim to provide a new matrix description of the celebrated Wiener index. In fact, we introduce the Wiener–Hosoya matrix of G, which is defined as the n×n matrix whose (i,j)-entry is equal to σi2di+σj2dj if vi and vj are adjacent and 0 otherwise. Some properties, including upper and lower bounds for the eigenvalues of the Wiener–Hosoya matrix are obtained and the extremal cases are described. Further, we introduce the energy of this matrix.

List Colourings of Regular Hypergraphs

2012 ◽  
Vol 21 (1-2) ◽  
pp. 315-322 ◽  
Author(s):  
DAVID SAXTON ◽  
ANDREW THOMASON
Keyword(s):  
Chromatic Number ◽  
Uniform Hypergraph ◽  
List Chromatic Number

We show that the list chromatic number of a simpled-regularr-uniform hypergraph is at least (1/2rlog(2r2) +o(1)) logdifdis large.

Decomposing Complete 3-Uniform Hypergraph into 5-Cycles

2014 ◽  
Vol 672-674 ◽  
pp. 1935-1939
Author(s):  
Guan Ru Li ◽  
Yi Ming Lei ◽  
Jirimutu
Keyword(s):  
Positive Integer ◽  
Hamiltonian Cycles ◽  
Uniform Hypergraph ◽  
Design Algorithm ◽  
Uniform Hypergraphs ◽  
Edge Partition ◽  
Definition Of

About the Katona-Kierstead definition of a Hamiltonian cycles in a uniform hypergraph, a decomposition of complete k-uniform hypergraph Kn(k) into Hamiltonian cycles studied by Bailey-Stevens and Meszka-Rosa. For n≡2,4,5 (mod 6), we design algorithm for decomposing the complete 3-uniform hypergraphs into Hamiltonian cycles by using the method of edge-partition. A decomposition of Kn(3) into 5-cycles has been presented for all admissible n≤17, and for all n=4m +1, m is a positive integer. In general, the existence of a decomposition into 5-cycles remains open. In this paper, we use the method of edge-partition and cycle sequence proposed by Jirimutu and Wang. We find a decomposition of K20(3) into 5-cycles.

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