Uniform Hypergraphs Research Articles

A Measure for the Vulnerability of Uniform Hypergraph Networks: Scattering Number

The scattering number of a graph G is defined as s(G)=max{ω(G−X)−|X|:X⊂V(G),ω(G−X)>1}, where X is a cut set of G, and ω(G−X) denotes the number of components in G−X, which can be used to measure the vulnerability of network G. In this paper, we generalize this parameter to a hypergraph to measure the vulnerability of uniform hypergraph networks. Firstly, some bounds on the scattering number are given. Secondly, the relations of scattering number between a complete k-uniform hypergraph and complete bipartite k-uniform hypergraph are discussed.

Rainbow connection numbers of some classes of $ s $-overlapping $ r $-uniform hypertrees with size $ t $

<abstract><p>The rainbow connection concept was developed to determine the minimum number of passwords required to exchange encrypted information between two agents. If the information exchange involves divisions managing more than two agents, the rainbow connection concept can be extended to a hypergraph. In 2014, Carpentier et al. expanded the rainbow connection concept of graphs to hypergraphs. They implemented it on a minimally connected hypergraph, an $ r $-uniform complete hypergraph, an $ r $-uniform cycle hypergraph, and an $ r $-uniform complete multipartite hypergraph. However, they did not determine the rainbow connection numbers of hypertrees. A hypergraph $ \mathcal{H} $ is called a hypertree if there exists a host tree $ T $ such that each edge of $ \mathcal{H} $ induces a subtree in $ T $. Therefore, in this article, we consider the rainbow connection numbers of some classes of $ s $-overlapping $ r $-uniform hypertrees with size $ t $. For $ r\geq 2 $, $ 1\leq s < r $, and $ t\geq 1 $, an $ s $-overlapping $ r $-uniform hypertree with size $ t $ is an $ r $-uniform connected hypertree, with $ s $ being the maximum cardinality of the vertex set obtained from the intersection of each pair of edges. We provide the best lower bound of the rainbow connection number of a connected hypergraph. Then, we determine the rainbow connection numbers of six classes of $ s $-overlapping $ r $-uniform hypertrees with size $ t $.</p></abstract>

Sparse random hypergraphs: non-backtracking spectra and community detection

Abstract We consider the community detection problem in a sparse $q$-uniform hypergraph $G$, assuming that $G$ is generated according to the Hypergraph Stochastic Block Model (HSBM). We prove that a spectral method based on the non-backtracking operator for hypergraphs works with high probability down to the generalized Kesten–Stigum detection threshold conjectured by Angelini et al. (2015, Spectral detection on sparse hypergraphs. In: 2015 53rd Annual Allerton Conference on Communication, Control, and Computing (Allerton). IEEE, pp. 66–73). We characterize the spectrum of the non-backtracking operator for the sparse HSBM and provide an efficient dimension reduction procedure using the Ihara–Bass formula for hypergraphs. As a result, community detection for the sparse HSBM on $n$ vertices can be reduced to an eigenvector problem of a $2n\times 2n$ non-normal matrix constructed from the adjacency matrix and the degree matrix of the hypergraph. To the best of our knowledge, this is the first provable and efficient spectral algorithm that achieves the conjectured threshold for HSBMs with $r$ blocks generated according to a general symmetric probability tensor.

Community Detection in Censored Hypergraph

Community detection refers to the problem of clustering the nodes of a network (either graph or hypergrah) into groups. Various algorithms are available for community detection and all these methods apply to uncensored networks. In practice, a network may has censored (or missing) values and it is shown that censored values have non-negligible effect on the structural properties of a network. In this paper, we study community detection in censored $m$-uniform hypergraph from information-theoretic point of view. We derive the information-theoretic threshold for exact recovery of the community structure. Besides, we propose a polynomial-time algorithm to exactly recover the community structure up to the threshold. The proposed algorithm consists of a spectral algorithm plus a refinement step. It is also interesting to study whether a single spectral algorithm without refinement achieves the threshold. To this end, we also explore the semi-definite relaxation algorithm and analyze its performance.

The spectral radii on uniform tricyclic hypergraphs

The spectral radii on uniform tricyclic hypergraphs

Uplifting edges in higher-order networks: Spectral centralities for non-uniform hypergraphs

<p>Spectral analysis of networks states that many structural properties of graphs, such as the centrality of their nodes, are given in terms of their adjacency matrices. The natural extension of such spectral analysis to higher-order networks is strongly limited by the fact that a given hypergraph could have several different adjacency hypermatrices, and hence the results obtained so far are mainly restricted to the class of uniform hypergraphs, which leaves many real systems unattended. A new method for analyzing non-linear eigenvector-like centrality measures of non-uniform hypergraphs was presented in this paper that could be useful for studying properties of $ \mathcal{H} $-eigenvectors and $ \mathcal{Z} $-eigenvectors in the non-uniform case. In order to do so, a new operation——the <italic>uplift</italic>——was introduced, incorporating auxiliary nodes in the hypergraph to allow for a uniform-like analysis. We later argued why this was a mathematically sound operation, and we furthermore used it to classify a whole family of hypergraphs with unique Perron-like $ \mathcal{Z} $-eigenvectors. We supplemented the theoretical analysis with several examples and numerical simulations on synthetic and real datasets: On the latter, we find a clear improvement over the existing methods, specially in cases where there is a huge disparity between the structure at each order, and on the former, we find that regardless of the chosen uniformization scheme, the nodes were similarly ranked.</p>

On the eccentricity-based invariants of uniform hypergraphs

On the eccentricity-based invariants of uniform hypergraphs

Sombor index of uniform hypergraphs

<p>In 2021, Gutman proposed a topological index for graphs known as the Sombor index. In this paper, we obtain several upper and lower bounds of the Sombor index of uniform hypergraphs, including those of hypertrees. Furthermore, we present a Nordhaus-Gaddum type result for the Sombor index of uniform hypergraphs.</p>

Bounded quantifier depth spectrum for random uniform hypergraphs

The notion of spectrum for first-order properties introduced by J. Spencer for Erdős–Rényi random graph is considered in relation to random uniform hypergraphs. In this work we study the set of limit points of the spectrum for first-order formulae with bounded quantifier depth and obtain bounds for its maximum value. Moreover, we prove zero–one k-laws for the random uniform hypergraph and improve the bounds for the maximum value of the spectrum for first-order formulae with bounded quantifier depth. We obtain that the maximum value of the spectrum belongs to some two-element set.

Book free 3-uniform hypergraphs

A k-book in a hypergraph consists of k Berge triangles sharing a common edge. In this paper we prove that the number of the hyperedges in a k-book-free 3-uniform hypergraph on n vertices is at most n28(1+o(1)).

Threshold graphs maximise homomorphism densities

AbstractGiven a fixed graph $H$ and a constant $c \in [0,1]$ , we can ask what graphs $G$ with edge density $c$ asymptotically maximise the homomorphism density of $H$ in $G$ . For all $H$ for which this problem has been solved, the maximum is always asymptotically attained on one of two kinds of graphs: the quasi-star or the quasi-clique. We show that for any $H$ the maximising $G$ is asymptotically a threshold graph, while the quasi-clique and the quasi-star are the simplest threshold graphs, having only two parts. This result gives us a unified framework to derive a number of results on graph homomorphism maximisation, some of which were also found quite recently and independently using several different approaches. We show that there exist graphs $H$ and densities $c$ such that the optimising graph $G$ is neither the quasi-star nor the quasi-clique (Day and Sarkar, SIAM J. Discrete Math. 35(1), 294–306, 2021). We also show that for $c$ large enough all graphs $H$ maximise on the quasi-clique (Gerbner et al., J. Graph Theory 96(1), 34–43, 2021), and for any $c \in [0,1]$ the density of $K_{1,2}$ is always maximised on either the quasi-star or the quasi-clique (Ahlswede and Katona, Acta Math. Hung. 32(1–2), 97–120, 1978). Finally, we extend our results to uniform hypergraphs.

A note on rainbow-free colorings of uniform hypergraphs

A rainbow-free coloring of a k-uniform hypergraph H is a vertex-coloring which uses k colors but with the property that no edge of H attains all colors. Koerkamp and Z̆ivný showed that p=(k−1)(logn)/n is the threshold function for the existence of a rainbow-free coloring of the random k-uniform hypergraph Gk(n,p), and presented that the case when p is close to the threshold is open. In this paper, we give an answer to the question.

Spectrum of 3-uniform 6- and 9-cycle systems over [formula omitted

We consider edge decompositions of Kv(3)−I, the complete 3-uniform hypergraph of order v minus a set of v/3 mutually disjoint edges (1-factor). We prove that a decomposition into tight 6-cycles exists if and only if v≡0,3,6 (mod 12) and v≥6; and a decomposition into tight 9-cycles exists for all v≥9 divisible by 3. These results are complementary to the theorems of Akin et al. [Discrete Math. 345 (2022)] and Bunge et al. [Australas. J. Combin. 80 (2021)] who settled the case of Kv(3).

Hypergraph analysis based on a compatible tensor product structure

We propose a tensor product structure that is compatible with the hypergraph structure. We define the algebraic connectivity of the (m+1)-uniform hypergraph in this product, and prove the relationship with the vertex connectivity. We introduce some connectivity optimization problem into the hypergraph, and solve them with the algebraic connectivity. We introduce the Laplacian eigenmap algorithm to the hypergraph under our tensor product.

A special case of Vu’s conjecture: colouring nearly disjoint graphs of bounded maximum degree

AbstractA collection of graphs is nearly disjoint if every pair of them intersects in at most one vertex. We prove that if $G_1, \dots, G_m$ are nearly disjoint graphs of maximum degree at most $D$ , then the following holds. For every fixed $C$ , if each vertex $v \in \bigcup _{i=1}^m V(G_i)$ is contained in at most $C$ of the graphs $G_1, \dots, G_m$ , then the (list) chromatic number of $\bigcup _{i=1}^m G_i$ is at most $D + o(D)$ . This result confirms a special case of a conjecture of Vu and generalizes Kahn’s bound on the list chromatic index of linear uniform hypergraphs of bounded maximum degree. In fact, this result holds for the correspondence (or DP) chromatic number and thus implies a recent result of Molloy and Postle, and we derive this result from a more general list colouring result in the setting of ‘colour degrees’ that also implies a result of Reed and Sudakov.

Transversals in regular uniform hypergraphs

AbstractThe transversal number of a hypergraph is the minimum number of vertices that intersect every edge of . This notion of transversal is fundamental in hypergraph theory and has been studied a great deal in the literature. A hypergraph is ‐regular if every vertex of has degree , that is, every vertex of belongs to exactly edges. Further, is ‐uniform if every edge of has size , and so every edge of is a ‐element subset of . For and , let be the class of all ‐regular ‐uniform hypergraphs of order . In this paper we study the problem posed by Tuza to determine or estimate the best possible constants (which depend only on and ) for each and , such that for all . These constants are given by , where the supremum is taken over all . Tuza presented closed formulas when or , and showed that for all , for even, and for odd. We conjecture that for all and that for all . We show that both these conjectures hold for . Moreover we show, for example, that and . We show that for every and for sufficiently large and , the growth of is given by .

The Degree and Codegree Threshold for Linear Triangle Covering in 3-Graphs

Given two $k$-uniform hypergraphs $F$ and $G$, we say that $G$ has an $F$-covering if every vertex in $G$ is contained in a copy of $F$. For $1\le i \le k-1$, let $c_i(n,F)$ be the least integer such that every $n$-vertex $k$-uniform hypergraph $G$ with $\delta_i(G)> c_i(n,F)$ has an $F$-covering. The covering problem has been systematically studied by Falgas-Ravry and Zhao [Codegree thresholds for covering 3-uniform hypergraphs, [SIAM J. Discrete Math., 2016]. Last year, Falgas-Ravry, Markström, and Zhao [Triangle-degrees in graphs and tetrahedron coverings in 3-graphs, Combinatorics, Probability and Computing, 2021] asymptotically determined $c_1(n, F)$ when $F$ is the generalized triangle. In this note, we give the exact value of $c_2(n, F)$ and asymptotically determine $c_1(n, F)$ when $F$ is the linear triangle $C_6^3$, where $C_6^3$ is the 3-uniform hypergraph with vertex set $\{v_1,v_2,v_3,v_4,v_5,v_6\}$ and edge set $\{v_1v_2v_3,v_3v_4v_5,v_5v_6v_1\}$.

On the Maximum $F_5$-Free Subhypergraphs of a Random Hypergraph

Denote by $F_5$ the $3$-uniform hypergraph on vertex set $\{1,2,3,4,5\}$ with hyperedges $\{123,124,345\}$. Balogh, Butterfield, Hu, and Lenz proved that if $p > K \log n /n$ for some large constant $K$, then every maximum $F_5$-free subhypergraph of $G^3(n,p)$ is tripartite with high probability, and showed that if $p_0 = 0.1\sqrt{\log n} /n$, then with high probability there exists a maximum $F_5$-free subhypergraph of $G^3(n,p_0)$ that is not tripartite. In this paper, we sharpen the upper bound to be best possible up to a constant factor. We prove that if $p > C \sqrt{\log n} /n $ for some large constant $C$, then every maximum $F_5$-free subhypergraph of $G^3(n, p)$ is tripartite with high probability.

Defective coloring of hypergraphs

AbstractWe prove that the vertices of every ‐uniform hypergraph with maximum degree may be colored with colors such that each vertex is in at most monochromatic edges. This result, which is best possible up to the value of the constant , generalizes the classical result of Erdős and Lovász who proved the case.

An improved bound for regular decompositions of 3-uniform hypergraphs of bounded VC2-dimension

An improved bound for regular decompositions of 3-uniform hypergraphs of bounded VC2-dimension