scholarly journals Weak and Strong Versions of the 1-2-3 Conjecture for Uniform Hypergraphs

10.37236/5709 ◽  
2016 ◽  
Vol 23 (2) ◽  
Author(s):  
Patrick Bennett ◽  
Andrzej Dudek ◽  
Alan Frieze ◽  
Laars Helenius
Keyword(s):  
Weight Function ◽  
Lower Bounds ◽  
Uniform Hypergraph ◽  
Uniform Hypergraphs ◽  
Np Complete ◽  
Almost All

Given an $r$-uniform hypergraph $H=(V,E)$ and a weight function $\omega:E\to\{1,\dots,w\}$, a coloring of vertices of~$H$, induced by~$\omega$, is defined by $c(v) = \sum_{e\ni v} w(e)$ for all $v\in V$. If there exists such a coloring that is strong (that means in each edge no color appears more than once), then we say that $H$ is strongly $w$-weighted. Similarly, if the coloring is weak (that means there is no monochromatic edge), then we say that $H$ is weakly $w$-weighted. In this paper, we show that almost all 3 or 4-uniform hypergraphs are strongly 2-weighted (but not 1-weighted) and almost all $5$-uniform hypergraphs are either 1 or 2 strongly weighted (with a nontrivial distribution). Furthermore, for $r\ge 6$ we show that almost all $r$-uniform hypergraphs are strongly 1-weighted. We complement these results by showing that almost all 3-uniform hypergraphs are weakly 2-weighted but not 1-weighted and for $r\ge 4$ almost all $r$-uniform hypergraphs are weakly 1-weighted. These results extend a previous work of Addario-Berry, Dalal and Reed for graphs. We also prove general lower bounds and show that there are $r$-uniform hypergraphs which are not strongly $(r^2-r)$-weighted and not weakly 2-weighted. Finally, we show that determining whether a particular uniform hypergraph is strongly 2-weighted is NP-complete.

On the spectral radii and principal eigenvectors of uniform hypergraphs

2017 ◽  
Vol 09 (04) ◽  
pp. 1750048 ◽  
Author(s):  
Xuelian Si ◽  
Xiying Yuan
Keyword(s):  
Spectral Radius ◽  
Lower Bounds ◽  
Uniform Hypergraph ◽  
Principal Eigenvector ◽  
Uniform Hypergraphs ◽  
Positive Eigenvector ◽  
Spectral Radius Formula

Let [Formula: see text] be a connected [Formula: see text]-uniform hypergraph. The unique positive eigenvector [Formula: see text] with [Formula: see text] corresponding to spectral radius [Formula: see text] is called the principal eigenvector of [Formula: see text]. In this paper, we present some lower bounds for the spectral radius [Formula: see text] and investigate the bounds of entries of the principal eigenvector of [Formula: see text].

scholarly journals Sharp Lower Bounds on the Spectral Radius of Uniform Hypergraphs Concerning Degrees

10.37236/6644 ◽  
2018 ◽  
Vol 25 (2) ◽  
Author(s):  
Liying Kang ◽  
Lele Liu ◽  
Erfang Shan
Keyword(s):  
Lower Bound ◽  
Spectral Radius ◽  
Linear Algebra ◽  
Lower Bounds ◽  
Uniform Hypergraph ◽  
Uniform Hypergraphs ◽  
Signless Laplacian Tensor ◽  
Signless Laplacian ◽  
Vertex Degrees

Let $\mathcal{A}(H)$ and $\mathcal{Q}(H)$ be the adjacency tensor and signless Laplacian tensor of an $r$-uniform hypergraph $H$. Denote by $\rho(H)$ and $\rho(\mathcal{Q}(H))$ the spectral radii of $\mathcal{A}(H)$ and $\mathcal{Q}(H)$, respectively. In this paper we present a  lower bound on $\rho(H)$ in terms of vertex degrees and we characterize the extremal hypergraphs attaining the bound, which solves a problem posed by Nikiforov [Analytic methods for uniform hypergraphs, Linear Algebra Appl. 457 (2014) 455–535]. Also, we prove a lower bound on $\rho(\mathcal{Q}(H))$ concerning degrees and give a characterization of the extremal hypergraphs attaining the bound.

scholarly journals The (signless Laplacian) spectral radius (of subgraphs) of uniform hypergraphs

Filomat ◽  
2019 ◽  
Vol 33 (15) ◽  
pp. 4733-4745 ◽  
Author(s):  
Cunxiang Duan ◽  
Ligong Wang ◽  
Peng Xiao ◽  
Xihe Li
Keyword(s):  
Spectral Radius ◽  
Lower Bounds ◽  
Maximum Degree ◽  
Uniform Hypergraph ◽  
Laplacian Spectral Radius ◽  
Signless Laplacian Spectral Radius ◽  
Uniform Hypergraphs ◽  
Signless Laplacian ◽  
Proper Subgraph

Let ?1(G) and q1(G) be the spectral radius and the signless Laplacian spectral radius of a kuniform hypergraph G, respectively. In this paper, we give the lower bounds of d-?1(H) and 2d-q1(H), where H is a proper subgraph of a f (-edge)-connected d-regular (linear) k-uniform hypergraph. Meanwhile, we also give the lower bounds of 2?-q1(G) and ?-?1(G), where G is a nonregular f (-edge)-connected (linear) k-uniform hypergraph with maximum degree ?.

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→ ∞.

Lower Bounds to the Eigenvalues in One-Dimensional Problems by a Shift in the Weight Function

1970 ◽  
Vol 37 (2) ◽  
pp. 267-270 ◽  
Author(s):  
D. Pnueli
Keyword(s):  
Lower Bound ◽  
Weight Function ◽  
Lower Bounds ◽  
Variational Formulation ◽  
Ritz Method ◽  
The Other ◽  
One Dimensional ◽  
Systematic Shift ◽  
The One ◽  
Detailed Computation

A method is presented to obtain both upper and lower bound to eigenvalues when a variational formulation of the problem exists. The method consists of a systematic shift in the weight function. A detailed procedure is offered for one-dimensional problems, which makes improvement of the bounds possible, and which involves the same order of detailed computation as the Rayleigh-Ritz method. The main contribution of this method is that it yields the “other bound;” i.e., the one which cannot be obtained by the Rayleigh-Ritz method.

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.

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