scholarly journals Upper Bounds on the Minimum Size of Hamilton Saturated Hypergraphs

10.37236/5252 ◽  
2016 ◽  
Vol 23 (4) ◽  
Author(s):  
Andrzej Ruciński ◽  
Andrzej Żak
Keyword(s):  
Upper Bound ◽  
Upper Bounds ◽  
Minimum Size ◽  
Uniform Hypergraph ◽  
Open Case ◽  
Vertex Ordering

For $1\leqslant \ell< k$,  an $\ell$-overlapping $k$-cycle is a $k$-uniform hypergraph in which, for some cyclic vertex ordering, every edge consists of $k$ consecutive vertices and every two consecutive edges share exactly $\ell$ vertices.A $k$-uniform hypergraph $H$ is $\ell$-Hamiltonian saturated if $H$ does not contain an $\ell$-overlapping Hamiltonian $k$-cycle but every hypergraph obtained from $H$ by adding one edge does contain such a cycle. Let $\mathrm{sat}(n,k,\ell)$ be the smallest number of edges in an $\ell$-Hamiltonian saturated $k$-uniform hypergraph on $n$ vertices. In the case of graphs Clark and Entringer showed in 1983 that $\mathrm{sat}(n,2,1)=\lceil \tfrac{3n}2\rceil$. The present authors proved that for $k\geqslant 3$ and $\ell=1$, as well as for all $0.8k\leqslant \ell\leq k-1$, $\mathrm{sat}(n,k,\ell)=\Theta(n^{\ell})$. In this paper we prove two upper bounds which cover the remaining range of $\ell$. The first, quite technical one, restricted to $\ell\geqslant\frac{k+1}2$, implies in particular that for $\ell=\tfrac23k$ and $\ell=\tfrac34k$ we have $\mathrm{sat}(n,k,\ell)=O(n^{\ell+1})$. Our main result provides an upper bound $\mathrm{sat}(n,k,\ell)=O(n^{\frac{k+\ell}2})$ valid for all $k$ and $\ell$. In the smallest open case we improve it further to $\mathrm{sat}(n,4,2)=O(n^{\frac{14}5})$.

Conflict-Free Colourings of Uniform Hypergraphs With Few Edges

2012 ◽  
Vol 21 (4) ◽  
pp. 611-622 ◽  
Author(s):  
A. KOSTOCHKA ◽  
M. KUMBHAT ◽  
T. ŁUCZAK
Keyword(s):  
Chromatic Number ◽  
Upper Bound ◽  
Upper Bounds ◽  
Uniform Hypergraph ◽  
Lower And Upper Bounds ◽  
Uniform Hypergraphs ◽  
Minimum Number

A colouring of the vertices of a hypergraph is called conflict-free if each edge e of contains a vertex whose colour does not repeat in e. The smallest number of colours required for such a colouring is called the conflict-free chromatic number of , and is denoted by χCF(). Pach and Tardos proved that for an (2r − 1)-uniform hypergraph with m edges, χCF() is at most of the order of rm1/r log m, for fixed r and large m. They also raised the question whether a similar upper bound holds for r-uniform hypergraphs. In this paper we show that this is not necessarily the case. Furthermore, we provide lower and upper bounds on the minimum number of edges of an r-uniform simple hypergraph that is not conflict-free k-colourable.

scholarly journals Hamilton Saturated Hypergraphs of Essentially Minimum Size

10.37236/2484 ◽  
2013 ◽  
Vol 20 (2) ◽  
Author(s):  
Andrzej Ruciński ◽  
Andrzej Żak
Keyword(s):  
Hamiltonian Cycle ◽  
Minimum Size ◽  
Uniform Hypergraph ◽  
Vertex Ordering

For $1\leq \ell< k$,  an $\ell$-overlapping cycle is a $k$-uniform hypergraph in which, for some cyclic vertex ordering, every edge consists of $k$ consecutive vertices and every two consecutive edges share exactly $\ell$ vertices. A $k$-uniform hypergraph $H$ is $\ell$-Hamiltonian saturated, $1\le \ell\le k-1$, if $H$ does not contain an $\ell$-overlapping Hamiltonian cycle $C^{(k)}_n(\ell)$ but every hypergraph obtained from $H$ by adding one more edge does contain $C^{(k)}_n(\ell)$. Let $sat(n,k,\ell)$ be the smallest number of edges in an $\ell$-Hamiltonian saturated $k$-uniform hypergraph on $n$ vertices. Clark and Entringer proved in 1983 that $sat(n,2,1)=\lceil \tfrac{3n}2\rceil$ and the second author showed recently that $sat(n,k,k-1)=\Theta(n^{k-1})$ for every $k\ge2$. In this paper we prove that $sat(n,k,\ell)=\Theta(n^{\ell})$ for $\ell=1$ as well as for all $k\ge5$ and $\ell\ge0.8k$.

scholarly journals On the Minimum Size of Hamilton Saturated Hypergraphs

10.37236/8414 ◽  
2020 ◽  
Vol 27 (4) ◽  
Author(s):  
Andrzej Ruciński ◽  
Andrzej Żak
Keyword(s):  
Minimum Size ◽  
Uniform Hypergraph ◽  
Vertex Ordering

For $1\leq \ell< k$,  an $\ell$-overlapping $k$-cycle is a $k$-uniform hypergraph in which, for some cyclic vertex ordering, every edge consists of $k$ consecutive vertices and every two consecutive edges share exactly $\ell$ vertices. A $k$-uniform hypergraph $H$ is $\ell$-hamiltonian saturated if $H$ does not contain an $\ell$-overlapping hamiltonian $k$-cycle but every hypergraph obtained from $H$ by adding one edge does contain such a cycle. Let sat$(N,k,\ell)$ be the smallest number of edges in an $\ell$-hamiltonian saturated $k$-uniform hypergraph on $N$ vertices. In the case of graphs Clark and Entringer showed in 1983 that sat$(N,2,1)=\lceil \tfrac{3N}2\rceil$. The present authors proved that for $k\geq 3$ and $\ell=1$, as well as for all $0.8k\leq \ell\leq k-1$, sat$(N,k,\ell)=\Theta(N^{\ell})$. Here we prove that sat$(N,2\ell,\ell)=\Theta\left(N^\ell\right)$.

scholarly journals Bandwidth of Graphs Resulting from the Edge Clique Covering Problem

10.37236/6900 ◽  
2018 ◽  
Vol 25 (4) ◽  
Author(s):  
Konrad Engel ◽  
Sebastian Hanisch
Keyword(s):  
Upper Bound ◽  
Covering Problem ◽  
Uniform Hypergraph ◽  
Asymptotic Value ◽  
Uniform Hypergraphs ◽  
Open Case ◽  
The Right ◽  
Weak Edge

Let $n,k,b$ be integers with $1 \le k-1 \le b \le n$ and let $G_{n,k,b}$ be the graph whose vertices are the $k$-element subsets $X$ of $\{0,\dots,n\}$ with $\mathrm{max}(X)-\mathrm{min}(X) \le b$ and where two such vertices $X,Y$ are joined by an edge if $\mathrm{max}(X \cup Y) - \mathrm{min}(X \cup Y) \le b$. These graphs are generated by applying a transformation to maximal $k$-uniform hypergraphs of bandwidth $b$ that is used to reduce the (weak) edge clique covering problem to a vertex clique covering problem. The bandwidth of $G_{n,k,b}$ is thus the largest possible bandwidth of any transformed $k$-uniform hypergraph of bandwidth $b$. For $b\geq \frac{n+k-1}{2}$, the exact bandwidth of these graphs is determined. Moreover, the bandwidth is determined asymptotically for $b=o(n)$ and for $b$ growing linearly in $n$ with a factor $\beta \in (0,1]$, where for one case only bounds could be found. It is conjectured that the upper bound of this open case is the right asymptotic value.

A note on improved upper bounds on the transversal number of hypergraphs

2019 ◽  
Vol 11 (01) ◽  
pp. 1950004
Author(s):  
Michael A. Henning ◽  
Nader Jafari Rad
Keyword(s):  
Upper Bound ◽  
Dominating Set ◽  
Domination Number ◽  
Minimum Size ◽  
Total Domination ◽  
Total Domination Number ◽  
Minimum Cardinality ◽  
Total Dominating Set ◽  
Isolated Vertex ◽  
Transversal Number

A subset [Formula: see text] of vertices in a hypergraph [Formula: see text] is a transversal if [Formula: see text] has a nonempty intersection with every edge of [Formula: see text]. The transversal number of [Formula: see text] is the minimum size of a transversal in [Formula: see text]. A subset [Formula: see text] of vertices in a graph [Formula: see text] with no isolated vertex, is a total dominating set if every vertex of [Formula: see text] is adjacent to a vertex of [Formula: see text]. The minimum cardinality of a total dominating set in [Formula: see text] is the total domination number of [Formula: see text]. In this paper, we obtain a new (improved) probabilistic upper bound for the transversal number of a hypergraph, and a new (improved) probabilistic upper bound for the total domination number of a graph.

Upper bounds on the energy dissipation in turbulent precession

1996 ◽  
Vol 321 ◽  
pp. 335-370 ◽  
Author(s):  
R. R. Kerswell
Keyword(s):  
Experimental Data ◽  
Energy Dissipation ◽  
Viscous Dissipation ◽  
Upper Bound ◽  
Dissipation Rate ◽  
Oblate Spheroid ◽  
Upper Bounds ◽  
Precession Rate ◽  
Long Cylinder

Rigorous upper bounds on the viscous dissipation rate are identified for two commonly studied precessing fluid-filled configurations: an oblate spheroid and a long cylinder. The latter represents an interesting new application of the upper-bounding techniques developed by Howard and Busse. A novel ‘background’ method recently introduced by Doering & Constantin is also used to deduce in both instances an upper bound which is independent of the fluid's viscosity and the forcing precession rate. Experimental data provide some evidence that the observed viscous dissipation rate mirrors this behaviour at sufficiently high precessional forcing. Implications are then discussed for the Earth's precessional response.

scholarly journals The Essential Dimension of Stacks of Parabolic Vector Bundles over Curves

2012 ◽  
Vol 10 (3) ◽  
pp. 455-488 ◽  
Author(s):  
Indranil Biswas ◽  
Ajneet Dhillon ◽  
Nicole Lemire
Keyword(s):  
Lower Bounds ◽  
Upper Bound ◽  
Vector Bundles ◽  
Upper Bounds ◽  
Essential Dimension ◽  
Parabolic Structure ◽  
Moduli Stack

AbstractWe find upper bounds on the essential dimension of the moduli stack of parabolic vector bundles over a curve. When there is no parabolic structure, we improve the known upper bound on the essential dimension of the usual moduli stack. Our calculations also give lower bounds on the essential dimension of the semistable locus inside the moduli stack of vector bundles of rank r and degree d without parabolic structure.

ances and covariances obtained from REML are normally distributed with expectation vector and variance-covariance matrix equal to the fol-low  ing, r  espectiv    ely,   When σˆ > 0.04, let νˆ = δˆ + σˆ + σˆ − 2ωˆ + σˆ − (1 + c (7.6) be an estimate for the (7.3) reference-scaled metric in accordance with FDA Guidance (2001) and using a REML UN model. Then (Patter-son, 2003; Patterson and Jones, 2002b), this estimate is asymptotically normally distributed and unbiased with E[νˆ ] = δ +σ − (1 + c and Var[νˆ ] = 4σ + l + 4l + (1 + c ) (l )+ 2l −2(1+c − 2(1+c +4(1+c −2(1+c . Similarly, for the constant-scaled metric, when σˆ ≤ 0.04, νˆ = δˆ + σˆ + σˆ − 2ωˆ + σˆ − σˆ − 0.04(c ) (7.7) E[νˆ ] = δ +σ − 0.04(c ) Var[νˆ ] = 4σ + l + 4l + 2l − 2l − 4l + 4l − 2l . The required asymptotic upper bound √ of the 90% confidence interval can √ then be calculated as νˆ + 1.645× V̂ ar[νˆ ] or νˆ + 1.645× V̂ ar[νˆ ], where the variances are obtained by ‘plugging in’ the estimated values of the variances and covariances obtained from SAS proc mixed into the formulae for Var[νˆ ] or Var[νˆ ]. The necessary SAS code to do this is given in Appendix B. The output reveals that σˆ = 0.0714 and the upper bound is−0.060 for log(AUC). For log(Cmax), σˆ = 0.1060 and the upper bound is −0.055. As both of these upper bounds are below zero, IBE can be claimed.

2003 ◽  
pp. 367-367
Keyword(s):  
Confidence Interval ◽  
Covariance Matrix ◽  
Upper Bound ◽  
Upper Bounds ◽  
Fda Guidance ◽  
Asymptotic Upper Bound ◽  
Variance Covariance Matrix ◽  
Normally Distributed

scholarly journals The Minimum Size of Signed Sumsets

10.37236/4881 ◽  
2015 ◽  
Vol 22 (2) ◽  
Author(s):  
Béla Bajnok ◽  
Ryan Matzke
Keyword(s):  
Abelian Group ◽  
Upper Bound ◽  
Finite Abelian Group ◽  
Minimum Size ◽  
Positive Integers

For a finite abelian group $G$ and positive integers $m$ and $h$, we let $$\rho(G, m, h) = \min \{ |hA| \; : \; A \subseteq G, |A|=m\}$$ and$$\rho_{\pm} (G, m, h) = \min \{ |h_{\pm} A| \; : \; A \subseteq G, |A|=m\},$$ where $hA$ and $h_{\pm} A$ denote the $h$-fold sumset and the $h$-fold signed sumset of $A$, respectively. The study of $\rho(G, m, h)$ has a 200-year-old history and is now known for all $G$, $m$, and $h$. Here we prove that $\rho_{\pm}(G, m, h)$ equals $\rho (G, m, h)$ when $G$ is cyclic, and establish an upper bound for $\rho_{\pm} (G, m, h)$ that we believe gives the exact value for all $G$, $m$, and $h$.

Models of arithmetic and upper bounds for arithmetic sets

1994 ◽  
Vol 59 (3) ◽  
pp. 977-983 ◽  
Author(s):  
Alistair H. Lachlan ◽  
Robert I. Soare
Keyword(s):  
Upper Bound ◽  
Upper Bounds ◽  
Models Of Arithmetic

AbstractWe settle a question in the literature about degrees of models of true arithmetic and upper bounds for the arithmetic sets. We prove that there is a model of true arithmetic whose degree is not a uniform upper bound for the arithmetic sets. The proof involves two forcing constructions.

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