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Generalization of graph theory
In mathematics, a hypergraph is a generalization of a graph in which an edge can join any number of vertices. In contrast, in an ordinary graph, an edge
Hypergraph
Set of hyperedges where every pair is disjoint
In graph theory, a matching in a hypergraph is a set of hyperedges, in which every two hyperedges are disjoint. It is an extension of the notion of matching
Matching_in_hypergraphs
theory, a balanced hypergraph is a hypergraph that has several properties analogous to that of a bipartite graph. Balanced hypergraphs were introduced by
Balanced_hypergraph
Generalizations in graph theory
theory, Hall-type theorems for hypergraphs are several generalizations of Hall's marriage theorem from graphs to hypergraphs. Such theorems were proved by
Hall-type theorems for hypergraphs
Hall-type_theorems_for_hypergraphs
In mathematics, a packing in a hypergraph is a partition of the set of the hypergraph's edges into a number of disjoint subsets such that no pair of edges
Packing_in_a_hypergraph
Set of hypergraph nodes to which every hyperedge is connected
In graph theory, a vertex cover in a hypergraph is a set of vertices, such that every hyperedge of the hypergraph contains at least one vertex of that
Vertex_cover_in_hypergraphs
artificial intelligence and operations research, constraint graphs and hypergraphs are used to represent relations among constraints in a constraint satisfaction
Constraint_graph
Conjecture in graph theory
relating the maximum matching size and the minimum transversal size in hypergraphs. This conjecture first appeared in 1971 in the Ph.D. thesis of J. R.
Ryser's_conjecture
is an algorithm that applies to hypergraphs. The algorithm takes as input a hypergraph and determines if the hypergraph is α-acyclic. If so, it computes
GYO_algorithm
Area of discrepancy theory
Discrepancy of hypergraphs is an area of discrepancy theory that studies the discrepancy of general set systems. In the classical setting, we aim at partitioning
Discrepancy_of_hypergraphs
American multinational information technology
Altair Engineering Inc. is an American multinational information technology company headquartered in Troy, Michigan, that provides software and cloud solutions
Altair_Engineering
British-American scientist (born 1959)
to reduce and explain all the laws of physics within a paradigm of a hypergraph that is transformed by minimal rewriting rules that obey the Church–Rosser
Stephen_Wolfram
Mathematical method in extremal graph theory
mathematics, the hypergraph regularity method is a powerful tool in extremal graph theory that refers to the combined application of the hypergraph regularity
Hypergraph_regularity_method
Graph divided into two independent sets
model a hypergraph in which U is the set of vertices of the hypergraph, V is the set of hyperedges, and E contains an edge from a hypergraph vertex v
Bipartite_graph
theory, there are two related properties of a hypergraph that are called its "width". Given a hypergraph H = (V, E), we say that a set K of edges pins
Width_of_a_hypergraph
Statement in mathematical combinatorics
induced Ramsey numbers to d-uniform hypergraphs by simply changing the word graph in the statement to hypergraph. Furthermore, we can define the multicolor
Ramsey's_theorem
In graph theory, the term bipartite hypergraph describes several related classes of hypergraphs, all of which are natural generalizations of a bipartite
Bipartite_hypergraph
Abstract simplicial complex describing a graph's cliques
independence complexes, flag complexes, Whitney complexes and conformal hypergraphs are closely related mathematical objects in graph theory and geometric
Clique_complex
Canadian mathematician
theory. He is renowned for his research on Turán's extremal problem for hypergraphs. He studied mathematics at McGill University, where he earned a Bachelor
Dominique_de_Caen
Theorem bounding chromatic number of symmetric graphs
The Symmetric hypergraph theorem is a theorem in combinatorics that puts an upper bound on the chromatic number of a graph (or hypergraph in general).
Symmetric_hypergraph_theorem
Property B is equivalent to 2-coloring the hypergraph described by the collection C {\displaystyle C} . A hypergraph with property B is also called 2-colorable
Property_B
Matrix that shows the relationship between two classes of objects
contrast, a hypergraph can have multiple vertices assigned to one edge; thus, a general matrix of non-negative integers describes a hypergraph. The incidence
Incidence_matrix
Hypergraph representing intervals on real number lines
In graph theory, a d-interval hypergraph is a kind of a hypergraph constructed using intervals of real lines. The parameter d is a positive integer. The
D-interval_hypergraph
Field of knowledge
includes counting configurations of geometric shapes. Graph theory and hypergraphs Coding theory, including error correcting codes and a part of cryptography
Mathematics
1979 conjecture in combinatorics
A hypergraph representing a family of union-closed sets. Vertices 1 and 2 (highlighted red and blue respectively) are present in over half the edges.
Union-closed_sets_conjecture
Theorem in graph theory
In graph theory, the hypergraph removal lemma states that when a hypergraph contains few copies of a given sub-hypergraph, then all of the copies can be
Hypergraph_removal_lemma
generalization of graph bipartiteness testing to 3-uniform hypergraphs: it asks whether the vertices of a hypergraph can be colored with two colors so that no hyperedge
Not-all-equal 3-satisfiability
Not-all-equal_3-satisfiability
Topics referred to by the same term
envelope of lines determined by a support function Hedgehog (hypergraph), a hypergraph formed from a complete graph by adding another vertex to each
Hedgehog_(disambiguation)
Creating a new graph from an existing graph
mentioned in the above section on the algebraic approach to graph rewriting. Hypergraph grammars, including as more restrictive subclasses port graph grammars
Graph_rewriting
Set that intersects every one of a family of sets
application domains, with the input family of sets often being described as a hypergraph. In set theory, the axiom of choice is equivalent to the statement that
Transversal_(combinatorics)
Research field in deep learning
hierarchies. This approach leverages structures like simplicial complexes and hypergraphs to capture global dependencies and qualitative spatial properties, offering
Topological_deep_learning
Decomposition of a graph into hamiltonion cycles
for hypergraphs are in general much harder than for graphs. Unlike graphs, hypergraphs admit multiple non-equivalent notions of cycles (see Hypergraph cycles)
Hamiltonian_decomposition
This is a list of some of the more commonly known problems that are NP-complete when expressed as decision problems. As there are thousands of such problems
List_of_NP-complete_problems
Method in combinatorics
The method of (hypergraph) containers is a powerful tool that can help characterize the typical structure and/or answer extremal questions about families
Container_method
Graph representing edges of another graph
line graphs of line graphs, line graphs of multigraphs, line graphs of hypergraphs, and line graphs of weighted graphs. Given a graph G, its line graph
Line_graph
Knowledge organization system
software modules, individual files, and events, associations, representing hypergraph relationships between topics, and occurrences, representing information
Topic_map
Fewest graph edges whose removal breaks all cycles
for a k-uniform hypergraph. This formula is symmetric between vertices and edges which demonstrates a hypergraph and its dual hypergraph have the same cyclomatic
Cyclomatic_number
AI benchmark testbed for mathematical problem solving
open problem—of the "moderately interesting" rank—to be solved was in hypergraph theory: "A Constant-Factor Lower Bound For H (n)" by GPT-5.4. Longest
FrontierMath
Graph with multiple edges between two vertices
two nodes, these are different edges. A multigraph is different from a hypergraph, which is a graph in which an edge can connect any number of nodes, not
Multigraph
A classical approach to solve the Hypergraph bipartitioning problem is an iterative heuristic by Charles Fiduccia and Robert Mattheyses. This heuristic
Fiduccia–Mattheyses_algorithm
Natural number
26 and preceding 28. Including the null-motif, there are 27 distinct hypergraph motifs. There are exactly twenty-seven straight lines on a smooth cubic
27_(number)
Extremal graph theory bound on clique-free graph edges
3 {\displaystyle 3} -uniform hypergraph can have without containing the complete 3 {\displaystyle 3} -uniform hypergraph on 4 {\displaystyle 4} vertices
Turán's_theorem
Theorem in convex and algebraic geometry
A multi-hypergraph over a certain set V {\displaystyle V} is a multiset of subsets of V {\displaystyle V} (it is called "multi-hypergraph" since each
Gordan's_lemma
Type of directed hypergraph
directed hypergraph where each hyperedge is directed either to one particular vertex or away from one particular vertex. In a directed hypergraph, each hyperedge
BF-graph
Generalization of line graphs to hypergraphs
In graph theory, particularly in the theory of hypergraphs, the line graph of a hypergraph H, denoted L(H), is the graph whose vertex set is the set of
Line_graph_of_a_hypergraph
Family of sets where every disjoint subfamily has k or fewer sets
into a space with Helly dimension 1. A hypergraph is equivalent to a set-family. In hypergraphs terms, a hypergraph H = (V, E) has the Helly property if
Helly_family
Form taken by the network of interconnections of a circuit
hypergraph, the tentacles carry labels which are determined by the hyperedge's label. A conventional directed graph can be thought of as a hypergraph
Circuit_topology_(electrical)
Subset of a graph's vertices, including at least one endpoint of every edge
problem. Vertex cover problems have been generalized to hypergraphs, see Vertex cover in hypergraphs. Formally, a vertex cover V ′ {\displaystyle V'} of an
Vertex_cover
Problem of grouping into triples
bipartite matching (also known as 2-dimensional matching) to 3-partite hypergraphs, which consist of hyperedges each of which contains 3 vertices (instead
3-dimensional_matching
Conjecture about coloring graphs
hypergraph with n hyperedges, one may n-color the vertices such that each hyperedge has one vertex of each color. A simple hypergraph is a hypergraph
Erdős–Faber–Lovász_conjecture
Graph whose maximal clique hypergraph is a hypertree
chordal if the hypergraph of its maximal cliques is a hypertree. The name comes from the fact that a graph is chordal if and only if the hypergraph of its maximal
Dually_chordal_graph
Partition into subsets from a given family
In turn, the incidence matrix can be seen also as describing a hypergraph. The hypergraph includes one node for each element in X and one edge for each
Exact_cover
Generalization of graph coloring to the hypergraph
coloring is a generalization of the notion of graph coloring to hypergraphs. A hypergraph H has a vertex-set V and an edge-set E. Each edge is a subset
Conflict-free_coloring
plane (TPP), also known as a dual affine plane, is a special kind of a hypergraph or geometric configuration that is constructed in the following way. Take
Truncated_projective_plane
Edge-colored graph matching where all edges have distinct colors
of edges. An r-uniform hypergraph is a set of hyperedges each of which contains exactly r vertices (so a 2-uniform hypergraph is a just a graph without
Rainbow_matching
Conjecture in probabilistic combinatorics
site percolation, hypergraphs, and directed graphs. The original disproof is based on a small counterexample in the 3-uniform hypergraph case with 10 vertices;
Bunkbed_conjecture
Subdivision of vertices into disjoint sets
bicriteria-approximation or resource augmentation approaches. A common extension is to hypergraphs, where an edge can connect more than two vertices. A hyperedge is not
Graph_partition
definition of cutset for hypergraphs: a cycle hypercutset of a hypergraph is a set of edges (rather than vertices) that makes the hypergraph acyclic when all
Decomposition method (constraint satisfaction)
Decomposition_method_(constraint_satisfaction)
structure. The notion of laminarity can be applied to hypergraphs to define "laminar hypergraphs" as those whose set of hyperedges forms a laminar set
Laminar_set_family
Upper bound on intersecting set families
can be formulated as part of the theory of hypergraphs. A family of sets may also be called a hypergraph, and when all the sets (which are called "hyperedges"
Erdős–Ko–Rado_theorem
Lubell–Yamamoto–Meshalkin inequality. They may also be described in the language of hypergraphs rather than set families, where they are called clutters. The number
Sperner_family
Mathematical models of strategic interactions
ISSN 0012-9682. Vannucci, Stefano (15 April 2025). "Von Neumann–Morgenstern Hypergraphs". Games. 16 (2): 17. doi:10.3390/g16020017. ISSN 2073-4336. This article
Game_theory
problems can also be formulated as constructing the transversal hypergraph of a given hypergraph, of listing all minimal hitting sets of a family of sets, or
Monotone_dualization
Number of edges touching a vertex in a graph
field of graph enumeration. More generally, the degree sequence of a hypergraph is the non-increasing sequence of its vertex degrees. A sequence is k
Degree_(graph_theory)
Any collection of sets, or subsets of a set
family of subsets of a finite set S {\displaystyle S} is also called a hypergraph. The subject of extremal set theory concerns the largest and smallest
Family_of_sets
Long dense subsets of the integers contain arbitrarily large arithmetic progressions
Arithmetic combinatorics Szemerédi regularity lemma Van der Waerden's theorem Hypergraph removal lemma § Proof of Szemerédi's theorem Erdős, Paul; Turán, Paul
Szemerédi's_theorem
Hungarian mathematician (born 1965)
of Sciences. His research concerns Hamiltonian cycles in graphs and hypergraphs, graph toughness, factors, and graph pebbling. In 2026 he was awarded
Gyula_Y._Katona
Canadian mathematician
research accomplishments include results on the Szemerédi regularity lemma, hypergraph generalizations of Hall's marriage theorem (see Haxell's matching theorem)
Penny_Haxell
File format for graphs
structure constellations including directed, undirected, mixed graphs, hypergraphs, and application-specific attributes. A GraphML file consists of an XML
GraphML
Computational complexity class
the intersection graph of disks, and determining the probability that a hypergraph becomes disconnected when some of its edges fail with given independent
Quasi-polynomial_time
American computer scientist and engineer
for Circuit partitioning: high-performance heuristic optimizations for hypergraph partitioning Placement: algorithms for finding ( x , y ) {\displaystyle
Igor_L._Markov
German mathematician (born 1977)
PhD in 2004 under the supervision of Vojtěch Rödl. His dissertation, on hypergraph generalizations of the Szemerédi regularity lemma, won the 2006 Richard
Mathias_Schacht
Largest independent set of paired elements
described as one of finding the largest Berge-acyclic sub-hypergraph of a 3-uniform hypergraph. A hypergraph is a structure analogous to a graph but allowing more
Matroid_parity_problem
In the mathematical theory of hypergraphs, a hedgehog is a 3-uniform hypergraph defined from an integer parameter t {\displaystyle t} . It has t + ( t
Hedgehog_(hypergraph)
Israeli mathematician
the first open case (that of 3-uniform hypergraphs) of a famous conjecture by Ryser: in a 3-partite hypergraph the ratio between the covering number and
Ron_Aharoni
Graph where every edge is in one triangle
linear graph form the hyperedges of a triangle-free 3-uniform linear hypergraph, and they form the blocks of certain partial Steiner triple systems; and
Locally_linear_graph
Probability theorem on no events occurring
Lovász and Paul Erdős in the article Problems and results on 3-chromatic hypergraphs and some related questions. For other versions, see Alon & Spencer (2000)
Lovász_local_lemma
Describing a family of graphs by excluding certain (sub)graphs
graph characterization is a method of specifying a family of graph, or hypergraph, structures, by specifying substructures that are forbidden to exist within
Forbidden graph characterization
Forbidden_graph_characterization
Generalization of tree graphs to hypergraphs
In the mathematical field of graph theory, a hypergraph H is called a hypertree if it admits a host graph T such that T is a tree. In other words, H is
Hypertree
Concept in projective geometry
geometry. One can define a blocking set of a hypergraph as a set that meets all edges of the hypergraph. In a finite projective plane π of order n, a
Blocking_set
Lighting technique
lighting setup Diyphotography.net Owen, G. Scott (2 November 1998). "Lighting for film and video". HyperGraph. ACM SIGGRAPH. Retrieved 28 October 2026.
Three-point_lighting
Context dependence in quantum measurements
understand contextuality, from the perspective of sheaf theory, graph theory, hypergraphs, algebraic topology, and probabilistic couplings. Nonlocality, in the
Quantum_contextuality
Generalized sphere of dimension n (mathematics)
S2CID 119297359. Meshulam, Roy (2001-01-01). "The Clique Complex and Hypergraph Matching". Combinatorica. 21 (1): 89–94. doi:10.1007/s004930170006. ISSN 1439-6912
N-sphere
German discrete mathematician
of Economics. Her research involves graph theory, including graph and hypergraph packing problems, random graphs and random subgraphs, and the relations
Julia_Böttcher
On coloring infinite graphs
directly to hypergraph coloring problems, where one requires that each hyperedge have vertices of more than one color. As for graphs, a hypergraph has a k
De Bruijn–Erdős theorem (graph theory)
De_Bruijn–Erdős_theorem_(graph_theory)
Graph partition into regular subgraphs
different notions of regularity and apply to other mathematical objects like hypergraphs. To state Szemerédi's regularity lemma formally, we must formalize what
Szemerédi_regularity_lemma
Abstract mathematical system of two types of objects and a relation between them
Incidence structures use geometric terminology, but in graph theory they are hypergraphs and in combinatorial design theory they are block designs. They are also
Incidence_structure
Hungarian mathematician
Baranyai is best known for his theorem on the decompositions of complete hypergraphs, which solved a long-standing open problem. In addition to his mathematical
Zsolt_Baranyai
Independent set in a graph
3-dimensional matching problem (3DM). The input to 3DM is a tripartite hypergraph (X + Y + Z, F), where X, Y, Z are vertex-sets of size m, and F is a set
Rainbow-independent_set
Area of discrete mathematics
problem, also called hitting set, can be described as a vertex cover in a hypergraph. Decomposition, defined as partitioning the edge set of a graph (with
Graph_theory
Subfield of computational complexity theory
(2019). "The average-case complexity of counting cliques in Erdős–Rényi hypergraphs". Proceedings of FOCS 2019. Carmosino, Marco L.; Gao, Jiawei; Impagliazzo
Fine_grained_complexity
products of graphs Walescki's theorem for hypergraphs: Do complete k {\displaystyle k} -uniform hypergraphs admit Hamiltonian decompositions into tight
List of unsolved problems in mathematics
List_of_unsolved_problems_in_mathematics
Israeli computer scientist (born 1960)
Processing Systems. 15. Zass, R; Shashua, A (2008). "Probabilistic graph and hypergraph matching". 2008 IEEE Conference on Computer Vision and Pattern Recognition
Amnon_Shashua
American computer scientist and AI researcher (born 1966)
at the Wayback Machine, Computerworld, 13 September 2007 "Patterns, Hypergraphs and Embodied General Intelligence", Ben Goertzel, WCCI Panel Discussion:
Ben_Goertzel
Topics referred to by the same term
a sequence Discrepancy theory in structural modelling Discrepancy of hypergraphs, an area of discrepancy theory Discrepancy (algebraic geometry) Discrepancy
Discrepancy
Matching which covers every node of the graph
Maximum-cardinality matching Perfect matching in high-degree hypergraphs Hall-type theorems for hypergraphs The unique perfect matching problem Alan Gibbons, Algorithmic
Perfect_matching
Category of positional games
specific hypergraphs are known on which the strategy stops working. In all conditions, k is the size of winning-sets (i.e., the game hypergraph is k-uniform)
Maker-Breaker_game
Theorem in graph theory
theorem on 3-term arithmetic progressions, and a generalization of it, the hypergraph removal lemma, can be used to prove Szemerédi's theorem. It also has applications
Graph_removal_lemma
Indian mathematician (born 1949)
research in block designs, projective planes, Intersection graphs of hypergraphs, and coding theory. He was a visiting professor at IIT Mumbai, University
Navin_M._Singhi
Topics referred to by the same term
conformal field theory Conformal fuel tanks on military aircraft Conformal hypergraph, in mathematics Conformal geometry, in mathematics Conformal group, in
Conformal
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