The poset of hypergraph quasirandomness

Chung and Graham began the systematic study of k‐uniform hypergraph quasirandom properties soon after the foundational results of Thomason and Chung‐Graham‐Wilson on quasirandom graphs. One feature that became apparent in the early work on k‐uniform hypergraph quasirandomness is that properties that are equivalent for graphs are not equivalent for hypergraphs, and thus hypergraphs enjoy a variety of inequivalent quasirandom properties. In the past two decades, there has been an intensive study of these disparate notions of quasirandomness for hypergraphs, and an open problem that has emerged is to determine the relationship between them. Our main result is to determine the poset of implications between these quasirandom properties. This answers a recent question of Chung and continues a project begun by Chung and Graham in their first paper on hypergraph quasirandomness in the early 1990's. © 2013 Wiley Periodicals, Inc. Random Struct. Alg., 46,762–800, 2015     

Constructing isospectral non‐isomorphic digraphs from hypergraphs

We explore the “oriented line graph” construction associated with a hypergraph, leading to a construction of pairs of strongly connected directed graphs whose adjacency operators have the same spectra. We give conditions on a hypergraph so that a hypergraph and its dual give rise to isospectral, but non‐isomorphic, directed graphs. The proof of isospectrality comes from an argument centered around hypergraph zeta functions as defined by Storm. To prove non‐isomorphism, we establish a Whitney‐type result by showing that the oriented line graphs are isomorphic if and only if the hypergraphs are. © 2009 Wiley Periodicals, Inc. J Graph Theory 63: 231–242, 2010     

An algorithmic hypergraph regularity lemma

Szemerédi 's Regularity Lemma is a powerful tool in graph theory. It asserts that all large graphs admit bounded partitions of their edge sets, most classes of which consist of uniformly distributed edges. The original proof of this result was nonconstructive, and a constructive proof was later given by Alon, Duke, Lefmann, Rödl, and Yuster. Szemerédi's Regularity Lemma was extended to hypergraphs by various authors. Frankl and Rödl gave one such extension in the case of 3‐uniform hypergraphs, which was later extended to k‐uniform hypergraphs by Rödl and Skokan. W.T. Gowers gave another such extension, using a different concept of regularity than that of Frankl, Rödl, and Skokan. Here, we give a constructive proof of a regularity lemma for hypergraphs.     

Graph Minors. XX. Wagner''s conjecture

In a previous paper (J. Combin. Theory 48 (1990) 255) we showed that for any infinite set of (finite) graphs drawn in a fixed surface, one of the graphs is isomorphic to a minor of another. In this paper we extend that result in two ways:

• we generalize from graphs to hypergraphs drawn in a fixed surface, in which each edge has two or three ends, and

• the edges of our hypergraphs are labelled from a well-quasi-order, and the minor relation is required to respect this order.

This result is another step in the proof of Wagner's conjecture, that for any infinite set of graphs, one is isomorphic to a minor of another.     

Note on isomorphic hypergraphs and some extensions of Whitney's theorem to families of sets

H. Whitney proved that, apart from a simple exeptional case, whenever the line graphs of two finite graphs are isomorphic then so are the graphs themselves. In this note (i) similar results are proved for finite hypergraphs, (ii) it is shown that certain extensions of Whitney's theorem to hypergraphs are false, (iii) a Whitney-type theorem is established for infinite hypergraphs.     

Hall’s and Kőnig’s theorem in graphs and hypergraphs

We investigate the relation between Hall’s theorem and K?nig’s theorem in graphs and hypergraphs. In particular, we characterize the graphs satisfying a deficiency version of Hall’s theorem, thereby showing that this class strictly contains all K?nig–Egerváry graphs. Furthermore, we give a generalization of Hall’s theorem to normal hypergraphs.     

Homomorphisms and amalgamation

We prove that for each finite core graph G, the class of all graphs admitting a homomorphism into G is a pseudo-amalgamation class, in the sense of Fraı̈ssé. This fact gives rise to a countably infinite universal pseudo-homogeneous graph which shares some of the properties of the infinite random graph. Our methods apply simultaneously to G-colourings in several classes of relational structures, such as the classes of directed graphs or hypergraphs.     

Cartesian products of directed graphs with loops

We show that every nontrivial finite or infinite connected directed graph with loops and at least one vertex without a loop is uniquely representable as a Cartesian or weak Cartesian product of prime graphs. For finite graphs the factorization can be computed in linear time and space.     

Constructing regular self‐complementary uniform hypergraphs

In this article, we examine the possible orders of t‐subset‐regular self‐complementary k‐uniform hypergraphs, which form examples of large sets of two isomorphic t‐designs. We reformulate Khosrovshahi and Tayfeh–Rezaie's necessary conditions on the order of these structures in terms of the binary representation of the rank k, and these conditions simplify to a more transparent relation between the order n and rank k in the case where k is a sum of consecutive powers of 2. Moreover, we present new constructions for 1‐subset‐regular self‐complementary uniform hypergraphs, and prove that these necessary conditions are sufficient for all k, in the case where t = 1. © 2011 Wiley Periodicals, Inc. J Combin Designs 19: 439‐454, 2011     

A Survey on Hypergraph Products

A surprising diversity of different products of hypergraphs have been discussed in the literature. Most of the hypergraph products can be viewed as generalizations of one of the four standard graph products. The most widely studied variant, the so-called square product, does not have this property, however. Here we survey the literature on hypergraph products with an emphasis on comparing the alternative generalizations of graph products and the relationships among them. In this context the so-called 2-sections and L2-sections are considered. These constructions are closely linked to related colored graph structures that seem to be a useful tool for the prime factor decompositions w.r.t. specific hypergraph products. We summarize the current knowledge on the propagation of hypergraph invariants under the different hypergraph multiplications. While the overwhelming majority of the material concerns finite (undirected) hypergraphs, the survey also covers a summary of the few results on products of infinite and directed hypergraphs.     

Counting acyclic hypergraphs

Acyclic hypergraphs are analogues of forests in graphs. They are very useful in the design of databases. The number of distinct acyclic uniform hypergraphs withn labeled vertices is studied. With the aid of the principle of inclusion-exclusion, two formulas are presented. One is the explicitformula for strict (d)-connected acyclic hypergraphs, the other is the recurrence formula for linear acyclic hypergraphs.     

On a question of Sós about 3‐uniform friendship hypergraphs

The well‐known Friendship Theorem states that if G is a graph in which every pair of vertices has exactly one common neighbor, then G has a single vertex joined to all others (a “universal friend”). V. Sós defined an analogous friendship property for 3‐uniform hypergraphs, and gave a construction satisfying the friendship property that has a universal friend. We present new 3‐uniform hypergraphs on 8, 16, and 32 vertices that satisfy the friendship property without containing a universal friend. We also prove that if n ≤ 10 and n ≠ 8, then there are no friendship hypergraphs on n vertices without a universal friend. These results were obtained by computer search using integer programming. © 2008 Wiley Periodicals, Inc. J Combin Designs 16: 253–261, 2008     

Oriented hypergraphs: Balanceability

An oriented hypergraph is an oriented incidence structure that extends the concepts of signed graphs, balanced hypergraphs, and balanced matrices. We introduce hypergraphic structures and techniques that generalize the circuit classification of the signed graphic frame matroid to any oriented hypergraphic incidence matrix via its locally-signed-graphic substructure. To achieve this, Camion's algorithm is applied to oriented hypergraphs to provide a generalization of reorientation sets and frustration that is only well-defined on balanceable oriented hypergraphs. A simple partial characterization of unbalanceable circuits extends the applications to representable matroids demonstrating that the difference between the Fano and non-Fano matroids is one of balance.     

On the Existence of Friendship Hypergraphs

A 3‐uniform friendship hypergraph is a 3‐uniform hypergraph in which, for all triples of vertices x, y, z there exists a unique vertex w, such that , and are edges in the hypergraph. Sós showed that such 3‐uniform friendship hypergraphs on n vertices exist with a so‐called universal friend if and only if a Steiner triple system, exists. Hartke and Vandenbussche used integer programming to search for 3‐uniform friendship hypergraphs without a universal friend and found one on 8, three nonisomorphic on 16 and one on 32 vertices. So far, these five hypergraphs are the only known 3‐uniform friendship hypergraphs. In this paper we construct an infinite family of 3‐uniform friendship hypergraphs on 2^k vertices and edges. We also construct 3‐uniform friendship hypergraphs on 20 and 28 vertices using a computer. Furthermore, we define r‐uniform friendship hypergraphs and state that the existence of those with a universal friend is dependent on the existence of a Steiner system, . As a result hereof, we know infinitely many 4‐uniform friendship hypergraphs with a universal friend. Finally we show how to construct a 4‐uniform friendship hypergraph on 9 vertices and with no universal friend.     

Cellular resolutions of cointerval ideals

Minimal cellular resolutions of the edge ideals of cointerval hypergraphs are constructed. This class of d-uniform hypergraphs coincides with the complements of interval graphs (for the case d?=?2), and strictly contains the class of ‘strongly stable’ hypergraphs corresponding to pure shifted simplicial complexes. The polyhedral complexes supporting the resolutions are described as certain spaces of directed graph homomorphisms, and are realized as subcomplexes of mixed subdivisions of the Minkowski sums of simplices. Resolutions of more general hypergraphs are obtained by considering decompositions into cointerval hypergraphs.     

Multicolor containers,extremal entropy,and counting

In breakthrough results, Saxton‐Thomason and Balogh‐Morris‐Samotij developed powerful theories of hypergraph containers. In this paper, we explore some consequences of these theories. We use a simple container theorem of Saxton‐Thomason and an entropy‐based framework to deduce container and counting theorems for hereditary properties of k‐colorings of very general objects, which include both vertex‐ and edge‐colorings of general hypergraph sequences as special cases. In the case of sequences of complete graphs, we further derive characterization and transference results for hereditary properties in terms of their stability families and extremal entropy. This covers within a unified framework a great variety of combinatorial structures, some of which had not previously been studied via containers: directed graphs, oriented graphs, tournaments, multigraphs with bounded multiplicity, and multicolored graphs among others. Similar results were recently and independently obtained by Terry.     

A hypergraph blow‐up lemma

We obtain a hypergraph generalisation of the graph blow‐up lemma proved by Komlós, Sarközy and Szemerédi, showing that hypergraphs with sufficient regularity and no atypical vertices behave as if they were complete for the purpose of embedding bounded degree hypergraphs. © 2011 Wiley Periodicals, Inc. Random Struct. Alg., 39, 275–376, 2011     

Operator Algebraic Quotient Structures Induced by Directed Graphs

Our result is about inclusions for (finite or infinite) countable directed graphs. In the proof, we use Free Probability Theory, groupoids, and algebras of operators (von Neumann algebras). We show that inclusions of directed graphs induce quotients for associated groupoid actions. With the use of operator thechniques, we then establish a duality between inclusions of graphs on the one hand and quotients of algebras on the other. Our main result is that each connected graph induces a quotient generated by a free group. This is a generalization of the notion of induced representations in the context of unitary representations of groups, i.e., the induction from the representations of a subgroup of an ambient group. The analogue is to systems of imprimitivity based on the homogeneous space. The parallel of this is the more general context of graphs (extending from groups to groupoids): We first prove that inclusions for connected graphs correspond to free group quotients, and we then build up the general case via connected components of given graphs.     

On the Size‐Ramsey Number of Hypergraphs

The size‐Ramsey number of a graph G is the minimum number of edges in a graph H such that every 2‐edge‐coloring of H yields a monochromatic copy of G. Size‐Ramsey numbers of graphs have been studied for almost 40 years with particular focus on the case of trees and bounded degree graphs. We initiate the study of size‐Ramsey numbers for k‐uniform hypergraphs. Analogous to the graph case, we consider the size‐Ramsey number of cliques, paths, trees, and bounded degree hypergraphs. Our results suggest that size‐Ramsey numbers for hypergraphs are extremely difficult to determine, and many open problems remain.     

图与超图中的彩色匹配综述

超图H=(V,E)是一个二元组(V,E),其中超边集E中的元素是点集V的非空子集.因此图是一种特殊的超图,超图也可以看作是一般图的推广.特别地,如果超边集E中的元素均是点集V的k元子集,则称该超图为k-一致的.通常情况下,为叙述简便,我们也会将超边简称为边.图(超图)中的匹配是指图(超图)中互不相交的边的集合.对于图(超图)中的彩色匹配,有两种定义方式:一为染色图(超图)中互不相交且颜色不同的边的集合;二为顶点集均为[n]的多个染色图(超图)所构成的集族中互不相交且颜色均不同的边的集合,且每条边均来自集族中不同的图(超图).现主要介绍了图与超图中关于彩色匹配的相关结果.