Asymptotic Enumeration of Sparse Uniform Linear Hypergraphs with Given Degrees

In 2018 Deza et al. proved the NP-completeness of deciding wether there exists a 3-uniform hypergraph compatible with a given degree sequence. A well known result of Erdos and Gallai (1960) shows that the same problem related to graphs can be solved in polynomial time. So, it becomes relevant to detect classes of uniform hypergraphs that are reconstructible in polynomial time. In particular, our study concerns 3-uniform hypergraphs that are defined in the NP-completeness proof of Deza et al. Those hypergraphs are constructed starting from a non-increasing sequence s of integers and have very interesting properties. In particular, they are unique, i.e., there do not exist two non isomorphic 3-uniform hypergraphs having the same degree sequence \(d_s\). This property makes us conjecture that the reconstruction of these hypergraphs from their degree sequences can be done in polynomial time. So, we first generalize the computation of the \(d_s\) degree sequences by Deza et al., and we show their uniqueness. We proceed by defining the equivalence classes of the integer sequences determining the same \(d_s\) and we define a (minimal) representative. Then, we find the asymptotic growth rate of the maximal element of the representatives in terms of the length of the sequence, with the aim of generating and then reconstructing them. Finally, we show an example of a unique 3-uniform hypergraph similar to those defined by Deza et al. that does not admit a generating integer sequence s. The existence of this hypergraph makes us conjecture an extended generating algorithm for the sequences of Deza et al. to include a much wider class of unique 3-uniform hypergraphs. Further studies could also include strategies for the identification and reconstruction of those new sequences and hypergraphs.

Enumeration of graphs with a heavy-tailed degree sequence

The signless Laplacian tensor and its H-eigenvalues for an even uniform hypergraph are introduced in this paper. Some fundamental properties of them for an even uniform hypergraph are obtained. In particular, the smallest and the largest H-eigenvalues of the signless Laplacian tensor for an even uniform hypergraph are discussed, and their relationships to hypergraph bipartition, minimum degree, and maximum degree are described. As an application, the bounds of the edge cut and the edge connectivity of the hypergraph involving the smallest and the largest H-eigenvalues are presented.

The expansion $G^+$ of a graph $G$ is the 3-uniform hypergraph obtained from $G$ by enlarging each edge of $G$ with a new vertex disjoint from $V(G)$ such that distinct edges are enlarged by distinct vertices. Let ${ex}_3(n,F)$ denote the maximum number of edges in a 3-uniform hypergraph with $n$ vertices not containing any copy of a 3-uniform hypergraph $F$. The study of ${ex}_3(n,G^+)$ includes some well-researched problems, including the case that $F$ consists of $k$ disjoint edges, $G$ is a triangle, $G$ is a path or cycle, and $G$ is a tree. In this paper we initiate a broader study of the behavior of ${ex}_3(n,G^+)$. Specifically, we show $ {ex}_3(n,K_{s,t}^+) = \Theta(n^{3 - 3/s})$ whenever $t > (s - 1)!$ and $s \geq 3$. One of the main open problems is to determine for which graphs $G$ the quantity ${ex}_3(n,G^+)$ is quadratic in $n$. We show that this occurs when $G$ is any bipartite graph with Turán number $o(n^{\varphi})$ where $\varphi = \frac{1 + \sqrt{5}}{2}$, and in particular this shows ${ex}_3(n,G^+) = O(n^2)$ when $G$ is the three-dimensional cube graph.

A nonnegative integer sequence is k-graphic if it is the degree sequence of a k-uniform simple hypergraph. The problem of deciding whether a given sequence pi is 3-graphic has recently been proved to be NP-complete, after years of studies. Thus, it acquires primary relevance to detect classes of degree sequences whose graphicality can be tested in polynomial time in order to restrict the NP-hard core of the problem and design algorithms that can also be useful in different research areas. Several necessary and few sufficient conditions for pi to be k-graphic, with kge 3, appear in the literature. Frosini et al. defined a polynomial time algorithm to reconstruct k-uniform hypergraphs having regular or almost regular degree sequences. Our study fits in this research line providing a combinatorial characterization of span-two sequences, i.e., sequences of the form pi =(d,ldots ,d,d-1,ldots ,d-1,d-2,ldots ,d-2), dge 2, which are degree sequences of some 3-uniform hypergraphs. Then, we define a polynomial time algorithm to reconstruct one of the related 3-uniform hypergraphs. Our results are likely to be easily generalized to k ge 4 and to other families of degree sequences having simple characterization, such as gap-free sequences.

Recent results establish for 2-spin antiferromagnetic systems that the computational complexity of approximating the partition function on graphs of maximum degree D undergoes a phase transition that coincides with the uniqueness phase transition on the infinite D-regular tree. For the ferromagnetic Potts model we investigate whether analogous hardness results hold. Goldberg and Jerrum showed that approximating the partition function of the ferromagnetic Potts model is at least as hard as approximating the number of independent sets in bipartite graphs (#BIS-hardness). We improve this hardness result by establishing it for bipartite graphs of maximum degree D. We first present a detailed picture for the phase diagram for the infinite D-regular tree, giving a refined picture of its first-order phase transition and establishing the critical temperature for the coexistence of the disordered and ordered phases. We then prove for all temperatures below this critical temperature that it is #BIS-hard to approximate the partition function on bipartite graphs of maximum degree D. As a corollary, it is #BIS-hard to approximate the number of k-colorings on bipartite graphs of maximum degree D when k <= D/(2 ln D). The #BIS-hardness result for the ferromagnetic Potts model uses random bipartite regular graphs as a gadget in the reduction. The analysis of these random graphs relies on recent connections between the maxima of the expectation of their partition function, attractive fixpoints of the associated tree recursions, and induced matrix norms. We extend these connections to random regular graphs for all ferromagnetic models and establish the Bethe prediction for every ferromagnetic spin system on random regular graphs. We also prove for the ferromagnetic Potts model that the Swendsen-Wang algorithm is torpidly mixing on random D-regular graphs at the critical temperature for large q.

We show that for each , in a density range extending up to, and slightly beyond, the threshold for a ‐factor, the copies of in the random graph are randomly distributed, in the (one‐sided) sense that the hypergraph that they form contains a copy of a binomial random hypergraph with almost exactly the right density. Thus Jeff Kahn's recent asymptotically sharp bound for the threshold in Shamir's hypergraph matching problem implies a corresponding bound for the threshold for to contain a ‐factor. The case is more difficult, and has been settled by Annika Heckel. We also prove a corresponding result for ‐factors in random ‐uniform hypergraphs, as well as (in some cases weaker) generalizations replacing by certain other (hyper)graphs.

Let MAJ_n denote the Boolean majority function of n input variables. In this paper, we study the construction of depth two circuits computing MAJ_n where each gate in a circuit computes MAJ_m for m < n. We first give an explicit construction of depth two MAJ_{floor[n/2]+2} o MAJ_{ = 7 such that n congruent 3 (mod 4) where MAJ_m and MAJ_{<= m} denote the majority gates that take m and at most m distinct inputs, respectively. A graph theoretic argument developed by Kulikov and Podolskii (STACS '17, Article No. 49) shows that there is no MAJ_{<= n-2} o MAJ_{n-2} circuit computing MAJ_n. Hence, our construction reveals that the use of a smaller fan-in gates at the bottom level is essential for the existence of such a circuit. Some computational results are also provided. We then show that the construction of depth two MAJ_m o MAJ_m circuits computing MAJ_n for m<n can be translated into the construction of a newly introduced version of bipartite expander graphs which we call a list expander. Intuitively, a list expander is a c-leftregular bipartite graph such that for a given d < c, every d-leftregular subgraph of the original graph has a certain expansion property. We formalize this connection and verify that, with high probability, a random bipartite graph is a list expander of certain parameters. However, the parameters obtained are not sufficient to give us a MAJ_{n-c} o MAJ_{n-c} circuit computing MAJ_n for a large constant c.

An edge‐coloring of a graph G with colors is called an interval t‐coloring if all colors are used, and the colors of edges incident to any vertex of G are distinct and form an interval of integers. In 1991, Erdős constructed a bipartite graph with 27 vertices and maximum degree 13 that has no interval coloring. Erdős's counterexample is the smallest (in a sense of maximum degree) known bipartite graph that is not interval colorable. On the other hand, in 1992, Hansen showed that all bipartite graphs with maximum degree at most 3 have an interval coloring. In this article, we give some methods for constructing of interval non‐edge‐colorable bipartite graphs. In particular, by these methods, we construct three bipartite graphs that have no interval coloring, contain 20, 19, 21 vertices and have maximum degree 11, 12, 13, respectively. This partially answers a question that arose in [T.R. Jensen, B. Toft, Graph coloring problems, Wiley Interscience Series in Discrete Mathematics and Optimization, 1995, p. 204]. We also consider similar problems for bipartite multigraphs.

The separation dimension of a hypergraph is the smallest natural number for which there is an embedding of into , such that any pair of disjoint edges is separated by some hyperplane normal to one of the axes. The perfect separation dimension further requires that any pair of disjoint edges is separated by the same amount of such (pairwise nonparallel) hyperplanes. While it is known that for any fixed , the separation dimension of any ‐vertex ‐graph is , the perfect separation dimension is much larger. In fact, no polynomial upper‐bound for the perfect separation dimension of ‐uniform hypergraphs is known. In our first result we essentially resolve the case , that is, graphs. We prove that the perfect separation dimension of is linear in , up to a small polylogarithmic factor. In fact, we prove it is at least and at most . Our second result proves that the perfect separation dimension of almost all graphs is also linear in , up to a logarithmic factor. This follows as a special case of a more general result showing that the perfect separation dimension of the random graph is w.h.p. for a wide range of values of , including all constant . Finally, we prove that significantly relaxing perfection to just requiring that any pair of disjoint edges of is separated the same number of times up to a difference of for some absolute constant , still requires the dimension to be . This is perhaps surprising as it is known that if we allow a difference of , then the dimension reduces to .

Motivated by an application in controllability we consider maximum matchings in random bipartite graphs G = (A, B). First we analyse Karp–Sipser's algorithm to determine the asymptotic size of maximum matchings in random bipartite graphs with a fixed degree distribution. We then allow an adversary to delete one edge adjacent to every vertex in A in the more restricted model where each vertex in A chooses d neighbours uniformly at random from B.

The independence number of a graph $G$, denoted $\alpha(G)$, is the maximum cardinality of an independent set of vertices in $G$. The independence number is one of the most fundamental and well-studied graph parameters. In this paper, we strengthen a result of Fajtlowicz [Combinatorica 4 (1984), 35-38] on the independence of a graph given its maximum degree and maximum clique size. As a consequence of our result we give bounds on the independence number and transversal number of $6$-uniform hypergraphs with maximum degree three. This gives support for a conjecture due to Tuza and Vestergaard [Discussiones Math. Graph Theory 22 (2002), 199-210] that if $H$ is a $3$-regular $6$-uniform hypergraph of order $n$, then $\tau(H) \le n/4$.

Using graph-theoretical techniques, we establish an inequality regarding the number of walks and closed walks in a graph. This inequality yields several upper bounds for the number of closed walks in a graph in terms of the number of vertices, number of edges, maximum degree, degree sequence, and the Zagreb indices of the graph. As applications, we also present some new upper bounds on the Estrada index for general graphs, bipartite graphs, trees and planar graphs, some of which improve the known results obtained by using the algebraic techniques.MSC:05C50, 94C15, 05C38.

Sidorenko’s conjecture states that, for all bipartite graphs $H$, quasirandom graphs contain asymptotically the minimum number of copies of $H$ taken over all graphs with the same order and edge density. While still open for graphs, the analogous statement is known to be false for hypergraphs. We show that there is some advantage in this, in that if Sidorenko’s conjecture does not hold for a particular $r$-partite $r$-uniform hypergraph $H$, then it is possible to improve the standard lower bound, coming from the probabilistic deletion method, for its extremal number $\textrm{ex}(n,H)$, the maximum number of edges in an $n$-vertex $H$-free $r$-uniform hypergraph. With this application in mind, we find a range of new counterexamples to the conjecture for hypergraphs, including all linear hypergraphs containing a loose triangle and all $3$-partite $3$-uniform tight cycles.

In this paper we focused our attention on a particular hybrid PN model called FOHPN, that is based on the fluidification of discrete PNs, and whose main feature is that the instantaneous firing speed of continuous transitions keeps constant during each macroperiod. In the first part of the paper we discuss in detail the advantages of fluidification, and provide a brief survey of the most important formalisms within the hybrid PN framework. Finally, we showed how FOHPNs can be efficiently used to model SC, and how interesting optimization problems can be solved via numerical simulation, by simply solving on-line a certain number of LPPs.