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Partition of a graph into spanning subgraphs
a k-factorization partitions the edges of the graph into disjoint k-factors. A graph G is said to be k-factorable if it admits a k-factorization. In particular
Graph_factorization
Function graph representing factorization
A factor graph is a bipartite graph representing the factorization of a function. In probability theory and its applications, factor graphs are used to
Factor_graph
Block design in combinatorial mathematics
matching with the factorization labels in turn. Similarly add three more blocks 12CDEF, 34CDEF, and 56CDEF, replacing the factorization labels by the corresponding
Steiner_system
Area of discrete mathematics
colorings of the graphs embedded on surfaces with arbitrary genus. Tait's reformulation generated a new class of problems, the factorization problems, particularly
Graph_theory
partition of the edges of the graph into factors; a k-factorization is a partition into k-factors. For instance a 1-factorization is an edge coloring with
Glossary_of_graph_theory
Operation in graph theory
the corresponding graphs cannot be decomposed. In this sense, the failure of unique factorization on (possibly disconnected) graphs is akin to the statement
Cartesian_product_of_graphs
\nu } . An immediate consequence of the factorization property is that morphisms in a k {\displaystyle k} -graph can be factored in multiple ways: there
K-graph_C*-algebra
Graph with a prism as its skeleton
of the largest possible number of 1-factorizations. A 1-factorization is a partition of the edge set of the graph into three perfect matchings, or equivalently
Prism_graph
Matching which covers every node of the graph
permutation matrix. A perfect matching is also called a 1-factor; see Graph factorization for an explanation of this term. In some literature, the term complete
Perfect_matching
Decomposition of a graph into hamiltonion cycles
undirected graphs and for directed graphs. In the undirected case a Hamiltonian decomposition can also be described as a 2-factorization of the graph such that
Hamiltonian_decomposition
k} -regular graph with 2 n {\displaystyle 2n} vertices is 1-factorable. The perfect 1-factorization conjecture that every complete graph on an even number
List of unsolved problems in mathematics
List_of_unsolved_problems_in_mathematics
true, would have numerous implications in graph theory, including the 1-factorization conjecture. For graphs in which Δ ≥ n / 3 {\displaystyle \Delta \geq
Overfull_graph
Type of matrix factorization
an LDU factorization (with all diagonal entries of L and U equal to 1), then the factorization is unique. In that case, the LU factorization is also
LU_decomposition
"GraphLab: Collaborative filtering library using matrix factorization methods". Archived from the original on 2016-12-20. Retrieved 2016-12-01. "GraphLab:
GraphLab
Representation of a matrix as a product
discipline of linear algebra, a matrix decomposition or matrix factorization is a factorization of a matrix into a product of matrices. There are many different
Matrix_decomposition
Accomplishments in factoring large integers
Integer factorization is the process of determining which prime numbers divide a given positive integer. Doing this quickly has applications in cryptography
Integer_factorization_records
Probabilistic model
the properties of factorization and independences, but they differ in the set of independences they can encode and the factorization of the distribution
Graphical_model
Aspect of mathematical group theory
and that between them include all 5 × 3 = 15 edges of the graph; this graph factorization can be done in 6 different ways. Consider a permutation of
Automorphisms of the symmetric and alternating groups
Automorphisms_of_the_symmetric_and_alternating_groups
Bijection between the vertex set of two graphs
In graph theory, an isomorphism of graphs G and H is a bijection between the vertex sets of G and H f : V ( G ) → V ( H ) {\displaystyle f\colon V(G)\to
Graph_isomorphism
Geometry problem on tiling by hypercubes
from there, into a problem in graph theory. Hajós (1949) first reformulated Keller's conjecture in terms of factorizations of abelian groups. He shows that
Keller's_conjecture
Concept in numerical linear algebra
algebra, an incomplete LU factorization (abbreviated as ILU) of a matrix is a sparse approximation of the LU factorization often used as a preconditioner
Incomplete_LU_factorization
Theorem in graph theory
same technique used for the 4-regular graph yields a factorization of a 2 k {\displaystyle 2k} -regular graph into two k {\displaystyle k} -factors.
2-factor_theorem
ax + by = c Integer factorization: breaking an integer into its prime factors Congruence of squares Dixon's algorithm Fermat's factorization method General
List_of_algorithms
Process of reducing the number of random variables under consideration
(LDA), canonical correlation analysis (CCA), or non-negative matrix factorization (NMF) techniques to pre-process the data, followed by clustering via
Dimensionality_reduction
Integer factorization algorithm
Pollard's rho algorithm is an algorithm for integer factorization. It was invented by John Pollard in 1975. It uses only a small amount of space, and
Pollard's_rho_algorithm
Type of mathematical expression
form, called factorization is, in general, too difficult to be done by hand-written computation. However, efficient polynomial factorization algorithms
Polynomial
Matrix decomposition method
In linear algebra, the Cholesky decomposition or Cholesky factorization (pronounced /ʃəˈlɛski/ shə-LES-kee) is a decomposition of a Hermitian, positive-definite
Cholesky_decomposition
Concept in machine learning
In 2009, the work of Sutskever introduced Bayesian Clustered Tensor Factorization to model relational concepts while reducing the parameter space. From
Tensor_(machine_learning)
Unsolved problem in computer science
quasi-polynomial time. The integer factorization problem is the computational problem of determining the prime factorization of a given integer. Phrased as
P_versus_NP_problem
Operation in graph theory
polynomial time algorithm for recognizing tensor product graphs and finding a factorization of any such graph. If either G or H is bipartite, then so is their
Tensor_product_of_graphs
Generalization of graph theory
design – Symmetric arrangement of finite sets Factor graph – Function graph representing factorization Greedoid – Set system used in greedy optimization
Hypergraph
Inherent difficulty of computational problems
perspectives on this. The integer factorization problem is the computational problem of determining the prime factorization of a given integer. Phrased as
Computational complexity theory
Computational_complexity_theory
Prime number of the form 2^n – 1
– Factorization of Mersenne numbers Mn (n up to 1280) Factorization of completely factored Mersenne numbers The Cunningham project, factorization of
Mersenne_prime
Maximal subgraph whose vertices can reach each other
Theorem 2, p. 59, and corollary, p. 65 Tutte, W. T. (1947), "The factorization of linear graphs", The Journal of the London Mathematical Society, 22 (2): 107–111
Component_(graph_theory)
Set of random variables
theorem) the graph is chordal (by equivalence to a Bayesian network) When such a factorization does exist, it is possible to construct a factor graph for the
Markov_random_field
Mathematical procedure
Matrix factorization is a class of collaborative filtering algorithms used in recommender systems. Matrix factorization algorithms work by decomposing
Matrix factorization (recommender systems)
Matrix_factorization_(recommender_systems)
Dimensionality reduction of graph-based semantic data objects [machine learning task]
Allen, Carl; Hospedales, Timothy M. (2019). "TuckER: Tensor Factorization for Knowledge Graph Completion". Proceedings of the 2019 Conference on Empirical
Knowledge_graph_embedding
1976 mathematics text
and graph coloring, a chapter on algebraic graph theory, and a final chapter on graph factorization. Appendices provide a brief update on graph history
Graph_Theory,_1736–1936
Complexity class
whether there is a polynomial-time algorithm for factorization, equivalently that integer factorization is in P, and hence this example is interesting as
Co-NP
Polynomial whose roots are the eigenvalues of a matrix
characteristic polynomial to zero. In spectral graph theory, the characteristic polynomial of a graph is the characteristic polynomial of its adjacency
Characteristic_polynomial
Topics referred to by the same term
without leaving a remainder Factorization, the decomposition of an object into a product of other objects Integer factorization, the process of breaking
Factor
Characterization of graphs with perfect matchings
Bondy & Murty (1976), pp. 76–78. Tutte, W. T. (1950). "The factorization of locally finite graphs". Canadian Journal of Mathematics. 2: 44–49. doi:10.4153/cjm-1950-005-2
Tutte's theorem on perfect matchings
Tutte's_theorem_on_perfect_matchings
Graph where all long cycles have a chord
In the mathematical area of graph theory, a chordal graph is one in which all cycles of four or more vertices have a chord, which is an edge that is not
Chordal_graph
Texas Instruments' response to a project to factorize cryptographic keys
te Riele (1999-08-26), New factorization record Archived 2021-06-24 at the Wayback Machine (announcement of factorization of RSA-155). Retrieved on 2008-03-10
Texas Instruments signing key controversy
Texas_Instruments_signing_key_controversy
Danish mathematician (1839–1910)
any such graph has a 2-factorization (2-factor theorem). (iii) Criteria for the existence of edge-separating factorizations of 4-regular graphs. (iv) The
Julius_Petersen
Dejter graph admits a 3-factorization into two copies of the Ljubljana graph, which is the third smallest existing semi-symmetric cubic graph of regular
Dejter_graph
List of unsolved computational problems
possible? Log-rank conjecture Hartmanis–Stearns conjecture Can integer factorization be done in polynomial time on a classical (non-quantum) computer? Can
List of unsolved problems in computer science
List_of_unsolved_problems_in_computer_science
Assignment of colors to edges of a graph
edges of the graph into perfect matchings, is the same thing as a k-edge-coloring of the graph. That is, a regular graph has a 1-factorization if and only
Edge_coloring
Clustering methods
Sibylle; Morik, Katharina (2018). The Relationship of DBSCAN to Matrix Factorization and Spectral Clustering (PDF). LWDA. pp. 330–334. Kannan, Ravi; Vempala
Spectral_clustering
is equivalent to a pair of Orthogonal 1-factorizations of K r + 1 {\textstyle K_{r+1}} , the complete graph on r + 1 {\textstyle r+1} vertices. Reading
Room_square
Natural number
In graph theory, all graphs with four or fewer vertices are planar, however, there is a graph with five vertices that is not: K5, the complete graph with
5
tree problem. Feedback vertex set Feedback arc set Graph coloring Graph homomorphism problem Graph partition into subgraphs of specific types (triangles
List_of_NP-complete_problems
function: polynomial of degree zero, graph is a horizontal straight line Linear function: First degree polynomial, graph is a straight line. Quadratic function:
List of mathematical functions
List_of_mathematical_functions
Problem in network theory
based on random walks. and matrix factorization have also been proposed With the advent of deep learning, several graph embedding based approaches for link
Link_prediction
Connectivity measure in graph theory
Dereniowski, Dariusz; Kubale, Marek (2004), "Cholesky Factorization of Matrices in Parallel and Ranking of Graphs", 5th International Conference on Parallel Processing
Cycle_rank
Number of times an object must be counted for making true a general formula
In prime factorization, the multiplicity of a prime factor is its p {\displaystyle p} -adic valuation. For example, the prime factorization of the integer
Multiplicity_(mathematics)
Graph with at most one cycle per component
In graph theory, a pseudoforest is an undirected graph in which every connected component has at most one cycle. That is, it is a system of vertices and
Pseudoforest
Natural number
Goldbach conjecture and, like it, remains unproven. Because of the factorization of 68 as 22 × (222 + 1), a 68-sided regular polygon may be constructed
68_(number)
p. 12.19. ISBN 1-58488-360-X. W.T. Tutte (April 1947). "The factorization of linear graphs" (PDF). J. London Math. Soc. 22 (2): 107–111. doi:10.1112/jlms/s1-22
Tutte_matrix
Theorem in optimal transport
convex functions. Brenier's theorem is closely related to the polar factorization theorem, also due to Yann Brenier, which decomposes a suitable vector
Brenier's_theorem
number of coprime integers less than an integer. Euler system Euler's factorization method Euler's Disk – a toy consisting of a circular disk that spins
List of topics named after Leonhard Euler
List_of_topics_named_after_Leonhard_Euler
Characterization of the size of a maximum matching in a graph
Springer-Verlag. p. 413. ISBN 3-540-44389-4. Tutte, W. T. (1947). "The factorization of linear graphs". Journal of the London Mathematical Society. Series 1. 22 (2):
Tutte–Berge_formula
Association of one output to each input
i is the canonical injection of f(X) into Y. This is the canonical factorization of f. "One-to-one" and "onto" are terms that were more common in the
Function_(mathematics)
Natural number
cubic 3-connected bipartite graph is Hamiltonian. Horton disproved the conjecture some years earlier with the Horton graph, but that was larger at 92 vertices
54_(number)
Matrix manipulation algorithm
derived a graph theoretic version of the algorithm where the factorization is only simulated, and this was named the minimum degree algorithm. The graph referred
Minimum_degree_algorithm
Hypothesis in computational complexity theory
two large primes n = p ⋅ q {\displaystyle n=p\cdot q} , the integer factorization problem is to find p {\displaystyle p} and q {\displaystyle q} (more
Computational hardness assumption
Computational_hardness_assumption
remain. Duval (1983) proposed an efficient algorithm involving the factorization of the string into its component Lyndon words, which runs in linear
Lexicographically minimal string rotation
Lexicographically_minimal_string_rotation
Algorithm for computing greatest common divisors
essential step in several integer factorization algorithms, such as Pollard's rho algorithm, Shor's algorithm, Dixon's factorization method and the Lenstra elliptic
Euclidean_algorithm
1979 classic textbook on computational complexity theory
are: Graph isomorphism This problem is known to be in NP, but it is unknown if it is NP-complete. Subgraph homeomorphism (for a fixed graph H) Graph genus
Computers_and_Intractability
Computational complexity of quantum algorithms
solvable by deterministic classical computers. For instance, integer factorization and the discrete logarithm problem are known to be in BQP and are suspected
Quantum_complexity_theory
Proving validity without revealing other data
mod m is known when m's factorization is not given. Moreover, all known NP proofs for this problem exhibit the prime factorization of m. This indicates that
Zero-knowledge_proof
Natural number
{\displaystyle N-1} is a prime number. The four-color theorem states that a planar graph (or, equivalently, a flat map of two-dimensional regions such as countries)
4
Natural number
90 → Cardinal seven Ordinal 7th (seventh) Numeral system septenary Factorization prime Prime 4th Divisors 1, 7 Greek numeral Ζ´ Roman numeral VII, vii
7
On finding a repeating loop in a sequence
possible. The classic example is Pollard's rho algorithm for integer factorization, which searches for a factor p of a given number n by looking for values
Cycle_detection
Family of stochastic optimization methods
bivariate factorizations was proposed in the EDA community, in which dependencies between pairs of variables could be modeled. A bivariate factorization can
Estimation of distribution algorithm
Estimation_of_distribution_algorithm
Mathematical problem
problems - A survey (open-access journal article) Zero-Sum Ramsey Theory: Graphs, Sequences and More (workshop homepage) Arie Bialostocki, "Zero-sum trees:
Zero-sum_problem
the polynomial and its derivative. The square-free factorization of a polynomial p is a factorization p = p 1 p 2 2 ⋯ p k k {\displaystyle p=p_{1}p_{2}^{2}\cdots
Polynomial_root-finding
Technique in machine learning
recognition Object detection Reinforcement learning: Game-playing Graph learning Matrix factorization Guo, Sheng; Huang, Weilin; Zhang, Haozhi; Zhuang, Chenfan;
Curriculum_learning
Complexity class
is a complexity class, standing for "Polynomial Parity Argument" (on a graph). Introduced by Christos Papadimitriou in 1994 (page 528), PPA is a subclass
PPA_(complexity)
Mathematical theorem
undirected graph G if and only if it is a Gibbs random field, that is, its density can be factorized over the cliques (or complete subgraphs) of the graph. The
Hammersley–Clifford_theorem
Probabilistic graphical representation of causal relationships
of variables and their conditional dependencies via a directed acyclic graph (DAG). While it is one of several forms of causal notation, causal networks
Bayesian_network
Topics referred to by the same term
Matrix decomposition, a factorization of a matrix into a product of matrices LU decomposition, a type of matrix factorization Permutation decomposition
Decomposition (disambiguation)
Decomposition_(disambiguation)
Argentine-born American mathematician
Invariants for 2-factorization and cycle systems, Triangles in 2-factorizations, Number of 4-cycles in 2-factorizations of complete graphs, Directed almost
Italo_Jose_Dejter
Polynomial equation of degree 3
straightforward computation allows verifying that the existence of this factorization is equivalent with Δ 0 = Δ 1 = 0. {\displaystyle \Delta _{0}=\Delta
Cubic_equation
Integer that is a perfect square modulo some integer
composite moduli whose prime factorization is known. In the case of a composite modulus with unknown prime factorization, the problem of identifying quadratic
Quadratic_residue
Feature of some statistical distributions
(Jeffrey) Hu, and Michael D. Smith, who first used a log-linear curve on an XY graph to describe the relationship between Amazon.com sales and sales ranking
Long_tail
Set of learning techniques in machine learning
include dictionary learning, independent component analysis, matrix factorization, and various forms of clustering. In self-supervised feature learning
Feature_learning
Discrete Fourier transform algorithm
factorize the DFT similarly to Cooley–Tukey but without the twiddle factors. The Rader–Brenner algorithm (1976) is a Cooley–Tukey-like factorization but
Fast_Fourier_transform
Method for finding stationary points of a function
applicable to certain types of equations, for example the Cholesky factorization and conjugate gradient will only work if f ″ ( x k ) {\displaystyle
Newton's method in optimization
Newton's_method_in_optimization
Hungarian mathematician (born 1948)
has been used in several practical applications, including polynomial factorization algorithms and cryptography. Donald Knuth named Lovász as one of his
László_Lovász
Polynomial function of degree 3
four parameters, their graph can have only very few shapes. In fact, the graph of a cubic function is always similar to the graph of a function of the form
Cubic_function
Data structure for Boolean functions
propositional directed acyclic graphs (PDAG). A Boolean function can be represented as a rooted, directed, acyclic graph, which consists of several (decision)
Binary_decision_diagram
Polynomial equation of degree two
given a quadratic equation in the form x2 + bx + c = 0, the sought factorization has the form (x + q)(x + s), and one has to find two numbers q and s
Quadratic_equation
Line of graphing calculators produced by Texas Instruments
te Riele (1999-08-26), New factorization record Archived 2004-12-31 at the Wayback Machine (announcement of factorization of RSA-155). Retrieved on 2008-03-10
TI-83_series
Machine learning algorithm
Models: Chordal Graphs, Decomposable Graphs, Junction Trees, and Factorizations" (PDF). Berkeley. Retrieved 14 December 2025. "Clique Graph". Retrieved 16
Junction_tree_algorithm
Natural number
which is more than twice 39, 39 is a Størmer number. The F26A graph is a symmetric graph with 39 edges. In Japanese wordplay and slang, through numeric
39_(number)
Decomposition of an integer as a sum of positive integers
different notion of rank Crank of a partition Dominance order Factorization Integer factorization Partition of a set Stars and bars (combinatorics) Plane partition
Integer_partition
Award for advancements in discrete mathematics
Kühn, Allan Lo, Deryk Osthus, and Andrew Treglown for Proof of the 1-factorization and Hamilton decomposition conjectures Jin-Yi Cai and Xi Chen for Complexity
Fulkerson_Prize
Field of mathematics
decompositions like the singular value decomposition, the QR factorization, the LU factorization, or the eigendecomposition, which can then be used to answer
Numerical_linear_algebra
Complexity class used to classify decision problems
problem in polynomial time. The decision problem version of the integer factorization problem: given integers n and k, is there a factor f with 1 < f < k
NP_(complexity)
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