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.

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.

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 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 $\mathrm{sat}(n,4,2)=O(n^{\frac{14}5})$.

scholarly journals Coefficient inequalities for a class of analytic functions associated with the lemniscate of Bernoulli

2018 ◽  
Vol 37 (4) ◽  
pp. 83-95
Author(s):  
Trailokya Panigrahi ◽  
Janusz Sokól
Keyword(s):  
Upper Bound ◽  
Analytic Functions ◽  
Hankel Determinant ◽  
Lemniscate Of Bernoulli ◽  
Toeplitz Determinant ◽  
Second Hankel Determinant ◽  
Coefficient Inequalities ◽  
The Right

In this paper, a new subclass of analytic functions ML_{\lambda}^{*}  associated with the right half of the lemniscate of Bernoulli is introduced. The sharp upper bound for the Fekete-Szego functional |a_{3}-\mu a_{2}^{2}|  for both real and complex \mu are considered. Further, the sharp upper bound to the second Hankel determinant |H_{2}(1)| for the function f in the class ML_{\lambda}^{*} using Toeplitz determinant is studied. Relevances of the main results are also briefly indicated.

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.

scholarly journals The Multiple-Orientability Thresholds for Random Hypergraphs

2015 ◽  
Vol 25 (6) ◽  
pp. 870-908 ◽  
Author(s):  
NIKOLAOS FOUNTOULAKIS ◽  
MEGHA KHOSLA ◽  
KONSTANTINOS PANAGIOTOU
Keyword(s):  
Load Balancing ◽  
Hard Disk ◽  
Maximum Load ◽  
Disk Arrays ◽  
Uniform Hypergraph ◽  
Cuckoo Hashing ◽  
Uniform Hypergraphs ◽  
Random Hypergraphs ◽  
Do So ◽  
Parallel Access

Ak-uniform hypergraphH= (V, E) is called ℓ-orientable if there is an assignment of each edgee∈Eto one of its verticesv∈esuch that no vertex is assigned more than ℓ edges. LetHn,m,kbe a hypergraph, drawn uniformly at random from the set of allk-uniform hypergraphs withnvertices andmedges. In this paper we establish the threshold for the ℓ-orientability ofHn,m,kfor allk⩾ 3 and ℓ ⩾ 2, that is, we determine a critical quantityc*k,ℓsuch that with probability 1 −o(1) the graphHn,cn,khas an ℓ-orientation ifc<c*k,ℓ, but fails to do so ifc>c*k,ℓ.Our result has various applications, including sharp load thresholds for cuckoo hashing, load balancing with guaranteed maximum load, and massive parallel access to hard disk arrays.

scholarly journals New Results on Degree Sequences of Uniform Hypergraphs

10.37236/3414 ◽  
2013 ◽  
Vol 20 (4) ◽  
Author(s):  
Sarah Behrens ◽  
Catherine Erbes ◽  
Michael Ferrara ◽  
Stephen G. Hartke ◽  
Benjamin Reiniger ◽  
...  
Keyword(s):  
Efficient Algorithm ◽  
Necessary Conditions ◽  
Degree Sequence ◽  
Sufficient Conditions ◽  
Degree Sequences ◽  
Uniform Hypergraph ◽  
Graphic Sequence ◽  
Uniform Hypergraphs ◽  
Open Question

A sequence of nonnegative integers is $k$-graphic if it is the degree sequence of a $k$-uniform hypergraph. The only known characterization of $k$-graphic sequences is due to Dewdney in 1975. As this characterization does not yield an efficient algorithm, it is a fundamental open question to determine a more practical characterization. While several necessary conditions appear in the literature, there are few conditions that imply a sequence is $k$-graphic. In light of this, we present sharp sufficient conditions for $k$-graphicality based on a sequence's length and degree sum.Kocay and Li gave a family of edge exchanges (an extension of 2-switches) that could be used to transform one realization of a 3-graphic sequence into any other realization. We extend their result to $k$-graphic sequences for all $k \geq 3$. Finally we give several applications of edge exchanges in hypergraphs, including generalizing a result of Busch et al. on packing graphic sequences.

scholarly journals Monochromatic Path and Cycle Partitions in Hypergraphs

10.37236/2631 ◽  
2013 ◽  
Vol 20 (1) ◽  
Author(s):  
András Gyárfás ◽  
Gábor N. Sárközy
Keyword(s):  
Uniform Hypergraph ◽  
Uniform Hypergraphs ◽  
Vertex Set

Here we address the problem to partition edge colored hypergraphs by monochromatic paths and cycles generalizing a well-known similar problem for graphs.We show that $r$-colored $r$-uniform complete hypergraphs can be partitioned into monochromatic Berge-paths of distinct colors. Also, apart from $2k-5$ vertices, $2$-colored $k$-uniform hypergraphs can be partitioned into two monochromatic loose paths.In general, we prove that in any $r$-coloring of a $k$-uniform hypergraph there is a partition of the vertex set intomonochromatic loose cycles such that their number depends only on $r$ and $k$.

scholarly journals Minimum-Weight Edge Discriminators in Hypergraphs

10.37236/3551 ◽  
2014 ◽  
Vol 21 (3) ◽  
Author(s):  
Bhaswar B. Bhattacharya ◽  
Sayantan Das ◽  
Shirshendu Ganguly
Keyword(s):  
Minimum Weight ◽  
Constant Factor ◽  
Uniform Hypergraph ◽  
Uniform Hypergraphs

In this paper we introduce the notion of minimum-weight edge-discriminators in hypergraphs, and study their various properties. For a hypergraph $\mathcal H=(\mathcal V, \mathscr E)$, a function $\lambda: \mathcal V\rightarrow \mathbb Z^{+}\cup\{0\}$ is said to be an edge-discriminator on $\mathcal H$ if $\sum_{v\in E_i}{\lambda(v)}>0$, for all hyperedges $E_i\in \mathscr E$, and $\sum_{v\in E_i}{\lambda(v)}\ne \sum_{v\in E_j}{\lambda(v)}$, for every two distinct hyperedges $E_i, E_j \in \mathscr E$. An optimal edge-discriminator on $\mathcal H$, to be denoted by $\lambda_\mathcal H$, is an edge-discriminator on $\mathcal H$ satisfying $\sum_{v\in \mathcal V}\lambda_\mathcal H (v)=\min_\lambda\sum_{v\in \mathcal V}{\lambda(v)}$, where the minimum is taken over all edge-discriminators on $\mathcal H$.  We prove that any hypergraph $\mathcal H=(\mathcal V, \mathscr E)$,  with $|\mathscr E|=m$, satisfies $\sum_{v\in \mathcal V} \lambda_\mathcal H(v)\leq m(m+1)/2$, and the equality holds if and only if the elements of $\mathscr E$ are mutually disjoint. For $r$-uniform hypergraphs $\mathcal H=(\mathcal V, \mathscr E)$, it follows from earlier results on Sidon sequences that $\sum_{v\in \mathcal V}\lambda_{\mathcal H}(v)\leq |\mathcal V|^{r+1}+o(|\mathcal V|^{r+1})$, and the bound is attained up to a constant factor by the complete $r$-uniform hypergraph. Finally, we show that no optimal edge-discriminator on any hypergraph $\mathcal H=(\mathcal V, \mathscr E)$, with $|\mathscr E|=m~(\geq 3)$, satisfies $\sum_{v\in \mathcal V} \lambda_\mathcal H (v)=m(m+1)/2-1$. This shows that all integer values between $m$ and $m(m+1)/2$ cannot be the weight of an optimal edge-discriminator of a hypergraph, and this raises many other interesting combinatorial questions.

Sign in / Sign up

Export Citation Format

Share Document