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Complements of perfect graphs are perfect
In graph theory, the perfect graph theorem of László Lovász (1972a, 1972b) states that an undirected graph is perfect if and only if its complement graph
Perfect_graph_theorem
Graph with tight clique-coloring relation
perfect graph theorem states that the complement graph of a perfect graph is also perfect. The strong perfect graph theorem characterizes the perfect
Perfect_graph
Perfect graphs have neither odd holes nor odd antiholes
In graph theory, the strong perfect graph theorem is a forbidden graph characterization of the perfect graphs as being exactly the graphs that have neither
Strong_perfect_graph_theorem
Graph representing edges of another graph
underlying graph from vertices into edges, and by Whitney's theorem the same translation can also be done in the other direction. Line graphs are claw-free
Line_graph
On bipartite matching and vertex cover
In the mathematical area of graph theory, Kőnig's theorem, proved by Dénes Kőnig (1931), describes an equivalence between the maximum matching problem
Kőnig's theorem (graph theory)
Kőnig's_theorem_(graph_theory)
On chains and antichains in partial orders
comparability graph is perfect: this is essentially just Mirsky's theorem, restated in graph-theoretic terms. By the perfect graph theorem of Lovász (1972)
Dilworth's_theorem
perfect 1. A perfect graph is a graph in which, in every induced subgraph, the chromatic number equals the clique number. The perfect graph theorem and
Glossary_of_graph_theory
Mathematical graph theorem
stated as follows: Petersen's Theorem. Every cubic, bridgeless graph contains a perfect matching. In other words, if a graph has exactly three edges at each
Petersen's_theorem
Graph divided into two independent sets
the strong perfect graph theorem, the perfect graphs have a forbidden graph characterization resembling that of bipartite graphs: a graph is bipartite
Bipartite_graph
Characterizes the height of any finite partially ordered set
complement graph of a comparability graph is perfect. The perfect graph theorem of Lovász (1972) states that the complements of perfect graphs are always
Mirsky's_theorem
Methodic assignment of colors to elements of a graph
graph introduced by Shannon. The conjecture remained unresolved for 40 years, until it was established as the celebrated strong perfect graph theorem
Graph_coloring
Describing a family of graphs by excluding certain (sub)graphs
forbidden graphs, the complete graph K5 and the complete bipartite graph K3,3. For Kuratowski's theorem, the notion of containment is that of graph homeomorphism
Forbidden graph characterization
Forbidden_graph_characterization
Area of discrete mathematics
results and conjectures concerning graph coloring are the following: Four-color theorem Strong perfect graph theorem Erdős–Faber–Lovász conjecture Total
Graph_theory
Result in combinatorics and graph theory
number of sets in the subset. The graph theoretic formulation answers whether a finite bipartite graph has a perfect matching—that is, a way to match each
Hall's_marriage_theorem
French mathematician (1926–2002)
if its complement is perfect, proven by László Lovász in 1972 and now known as the perfect graph theorem, and A graph is perfect if and only if neither
Claude_Berge
Matching which covers every node of the graph
In graph theory, a perfect matching in a graph is a matching that covers every vertex of the graph. More formally, given a graph G with edges E and vertices
Perfect_matching
Graph of chess rook moves
component of a decomposition of perfect graphs used to prove the strong perfect graph theorem, which characterizes all perfect graphs. The independence number
Rook's_graph
Characterization of graphs with perfect matchings
discipline of graph theory, the Tutte theorem, named after William Thomas Tutte, is a characterization of finite undirected graphs with perfect matchings
Tutte's theorem on perfect matchings
Tutte's_theorem_on_perfect_matchings
(Avraham Trahtman, 2007) Robertson–Seymour theorem (Neil Robertson, Paul Seymour, 2004) Strong perfect graph conjecture (Maria Chudnovsky, Neil Robertson
List of unsolved problems in mathematics
List_of_unsolved_problems_in_mathematics
Trail in which only the first and last vertices are equal
complement of a graph hole. Chordless cycles may be used to characterize perfect graphs: by the strong perfect graph theorem, a graph is perfect if and only
Cycle_(graph_theory)
Topics referred to by the same term
4-vertex-connected planar graphs Tutte's theorem on perfect matchings, a characterization of the graphs having perfect matchings Tutte's spring theorem, on the planarity
Tutte's_theorem
Mycielski's theorem (graph theory) Nicomachus's theorem (number theory) Ore's theorem (graph theory) Paley's theorem (algebra) Perfect graph theorem (graph theory)
List_of_theorems
Graph linking pairs of comparable elements in a partial order
is Dilworth's theorem; these facts, together with the perfect graph theorem can be used to prove Dilworth's theorem from Mirsky's theorem or vice versa
Comparability_graph
Graph which partitions into a clique and independent set
classes of perfect graphs from which all others can be formed in the proof by Chudnovsky et al. (2006) of the Strong Perfect Graph Theorem. If a graph is both
Split_graph
Generalizations in graph theory
and others. Hall's marriage theorem provides a condition guaranteeing that a bipartite graph (X + Y, E) admits a perfect matching, or - more generally
Hall-type theorems for hypergraphs
Hall-type_theorems_for_hypergraphs
Canadian-American mathematician (born 1938)
2006 for the Robertson–Seymour theorem, and in 2009 for his participation in the proof of the strong perfect graph theorem. He also won the Pólya Prize
Neil Robertson (mathematician)
Neil_Robertson_(mathematician)
Graph where all odd cycles of length ≥ 5 has 2+ chords
1976, long before the proof of the strong perfect graph theorem completely characterized the perfect graphs. The same result was independently discovered
Meyniel_graph
Graph with same nodes as but complementary connections to another
the complement of a perfect graph is also perfect is the perfect graph theorem of László Lovász. Cographs are defined as the graphs that can be built up
Complement_graph
Graph without four-vertex star subgraphs
subgraph. It is now known (the strong perfect graph theorem) that perfect graphs may be characterized as the graphs that do not have as induced subgraphs
Claw-free_graph
Mathematician and engineer
strong perfect graph theorem (with Neil Robertson, Paul Seymour, and Robin Thomas) characterizing perfect graphs as being exactly the graphs with no odd
Maria_Chudnovsky
Award for advancements in discrete mathematics
Neil Robertson, Paul Seymour, and Robin Thomas, for the strong perfect graph theorem. Daniel A. Spielman and Shang-Hua Teng, for smoothed analysis of
Fulkerson_Prize
Set of edges without common vertices
bipartite graphs. Hall's marriage theorem provides a characterization of bipartite graphs which have a perfect matching and Tutte's theorem on perfect matchings
Matching_(graph_theory)
Graph where every connected induced subgraph has a universal vertex
In graph theory, a trivially perfect graph is a graph with the property that in each of its induced subgraphs the size of the maximum independent set equals
Trivially_perfect_graph
Partition of a graph into spanning subgraphs
bipartite graph. Hall's marriage theorem can be used to show that a k-regular bipartite graph contains a perfect matching. One can then remove the perfect matching
Graph_factorization
al. (2006) to prove the strong perfect graph theorem that the Berge graphs are indeed the same as the perfect graphs. Chudnovsky et al. were unable to
Skew_partition
{\displaystyle n} -vertex graph. For AT-free graphs, the pathwidth equals the treewidth. The strong perfect graph theorem holds for AT-free graphs, as they are a
Asteroidal_triple-free_graph
Graph where all long cycles have a chord
induced cycle in the graph should have exactly three vertices. The chordal graphs may also be characterized as the graphs that have perfect elimination orderings
Chordal_graph
Characterization of the size of a maximum matching in a graph
of graph theory the Tutte–Berge formula is a characterization of the size of a maximum matching in a graph. It is a generalization of Tutte's theorem on
Tutte–Berge_formula
with several gigabytes of computer calculations. 2006 – the strong perfect graph theorem, by Maria Chudnovsky, Neil Robertson, Paul Seymour, and Robin Thomas
List of long mathematical proofs
List_of_long_mathematical_proofs
British mathematician
especially graph theory. He (with others) was responsible for important progress on regular matroids and totally unimodular matrices, the four colour theorem, linkless
Paul_Seymour_(mathematician)
Cubic graph with 10 vertices and 15 edges
bridgeless graph has a cycle-continuous mapping to the Petersen graph. More unsolved problems in mathematics In the mathematical field of graph theory, the
Petersen_graph
Mathematician (1962–2020)
the Hadwiger conjecture, and in 2009 for the proof of the strong perfect graph theorem. In 2011, he was awarded the Karel Janeček Foundation Neuron Prize
Robin_Thomas_(mathematician)
Graph made from a subset of another graph's nodes and their edges
to the strong perfect graph theorem, induced cycles and their complements play a critical role in the characterization of perfect graphs. Cliques and independent
Induced_subgraph
Bipartite graph where each node of 1st set is linked to all nodes of 2nd set
nonplanar graph contains either K3,3 or the complete graph K5 as a minor; this is Wagner's theorem. Every complete bipartite graph. Kn,n is a Moore graph and
Complete_bipartite_graph
Adjacent subset of an undirected graph
edges must contain a three-vertex clique. Ramsey's theorem states that every graph or its complement graph contains a clique with at least a logarithmic number
Clique_(graph_theory)
edge-coloring Perfect graph Ramsey's theorem Sperner's lemma Strong coloring Subcoloring Tait's conjecture Total coloring Uniquely colorable graph Path (graph theory)
List_of_graph_theory_topics
Upper bound on intersecting set families
Another analog of the theorem, for partitions of a set, includes as a special case the perfect matchings of a complete graph K n {\displaystyle K_{n}}
Erdős–Ko–Rado_theorem
Graph with all vertices of degree 3
on graph theory, that every cubic graph has an even number of vertices. Petersen's theorem states that every cubic bridgeless graph has a perfect matching
Cubic_graph
Sparse graph with strong connectivity
In graph theory, an expander graph is a sparse graph that has strong connectivity properties, quantified using vertex, edge or spectral expansion. Expander
Expander_graph
Moroccan mathematician
strong perfect graph theorem, for the graphs that have no bull graph as an induced subgraph.[B] Their work in this area introduced a type of graph decomposition
Najiba_Sbihi
Theorem in graph theory
mathematical discipline of graph theory, the 2-factor theorem, discovered by Julius Petersen, is one of the earliest works in graph theory. It can be stated
2-factor_theorem
Graph in which every two vertices are adjacent
characterizations of planar graphs: by Kuratowski's theorem, a graph is planar if and only if it contains neither K5 nor the complete bipartite graph K3,3 as a subdivision
Complete_graph
Maximal subgraph whose vertices can reach each other
components play a key role in Tutte's theorem on perfect matchings characterizing finite graphs that have perfect matchings and the associated Tutte–Berge
Component_(graph_theory)
Graph whose vertices correspond to combinations of a set of n elements
In graph theory, the Kneser graph K(n, k) (alternatively KGn,k) is the graph whose vertices correspond to the k-element subsets of a set of n elements
Kneser_graph
triangle-free graphs are bull-free graphs, since every bull contains a triangle. The strong perfect graph theorem was proven for bull-free graphs long before
Bull_graph
Refinement of perfect matching theorems
a concept in graph theory that is used to refine various theorems related to perfect matching in graphs, such as Hall's marriage theorem. This was first
Deficiency_(graph_theory)
Partition of a graph's nodes into cliques
to the weak perfect graph theorem, the complement of a perfect graph is also perfect. Therefore, the perfect graphs are also the graphs in which, for
Clique_cover
Any planar graph can be subdivided by removing a few vertices
In graph theory, the planar separator theorem is a form of isoperimetric inequality for planar graphs, that states that any planar graph can be split
Planar_separator_theorem
Graph formed by complementation and disjoint union
special cases of the distance-hereditary graphs, permutation graphs, comparability graphs, and perfect graphs. Any cograph may be constructed using the
Cograph
Graphs formed by a hypercube's edges and vertices
complete graph, and may be decomposed into two copies of Q n − 1 {\displaystyle Q_{n-1}} connected to each other by a perfect matching. Hypercube graphs should
Hypercube_graph
Graph path which is an induced subgraph
strong perfect graph theorem, the perfect graphs are the graphs with no odd hole and no odd antihole. The distance-hereditary graphs are the graphs in which
Induced_path
Theorem in combinatorics
number. The Dinitz theorem is also related to Rota's basis conjecture. Erdős, P.; Rubin, A. L.; Taylor, H. (1979). "Choosability in graphs". Proc. West Coast
Dinitz_conjecture
Graph representing intersections between given sets
intersection graph of unit disks in the plane. A circle graph is the intersection graph of a set of chords of a circle. The circle packing theorem states that
Intersection_graph
{\displaystyle Y} . By Hall's marriage theorem, the last condition implies the first one. In a general graph, the above conditions are not equivalent
Fractional_matching
Undirected graph with 14 vertices
distance-transitive graph (see the Foster census) and therefore distance regular. There are 24 perfect matchings in the Heawood graph; for each matching
Heawood_graph
Relationship between derivatives and integrals
first fundamental theorem may be interpreted as follows. Given a continuous function y = f ( x ) {\displaystyle y=f(x)} whose graph is plotted as a curve
Fundamental theorem of calculus
Fundamental_theorem_of_calculus
Balanced complete multipartite graph
Erdős–Stone theorem extends Turán's theorem by bounding the number of edges in a graph that does not have a fixed Turán graph as a subgraph. Via this theorem, similar
Turán_graph
Graph whose induced subgraphs preserve distance
graph is a perfect graph, more specifically a perfectly orderable graph and a Meyniel graph. Every distance-hereditary graph is also a parity graph,
Distance-hereditary_graph
diagonals Euclid–Euler theorem, characterizing even perfect numbers Euler's theorem, on modular exponentiation Euler's partition theorem relating the product
List of topics named after Leonhard Euler
List_of_topics_named_after_Leonhard_Euler
Non-crossing graph with vertices on outer face
outer face of the drawing. Outerplanar graphs may be characterized (analogously to Wagner's theorem for planar graphs) by the two forbidden minors K4 and
Outerplanar_graph
Set of hyperedges where every pair is disjoint
perfect if every vertex v in V is contained in exactly one hyperedge of M. This is the natural extension of the notion of perfect matching in a graph
Matching_in_hypergraphs
Partition of a graph whose components are reachable from all vertices
In the mathematical theory of directed graphs, a graph is said to be strongly connected if every vertex is reachable from every other vertex. The strongly
Strongly_connected_component
Hungarian-American mathematician
by C. Berge and P. Duchet (and which is independent of the perfect graph theorem). He settled the complexity of generating all maximal frequent and minimal
Endre_Boros
1976 mathematics text
Kuratowski's theorem, and Euler's polyhedral formula. There are three chapters on the four color theorem and graph coloring, a chapter on algebraic graph theory
Graph_Theory,_1736–1936
Partition of the vertices of a graph
In graph theory, the Gallai–Edmonds decomposition is a partition of the vertices of a graph into three subsets which provides information on the structure
Gallai–Edmonds_decomposition
Recursively splitting a graph into subsets of nodes
celebrated proof of the perfect graph theorem (Golumbic, 1980). For recognizing distance-hereditary graphs and circle graphs, a further generalization of modular
Modular_decomposition
Assigning directions to the edges of an undirected graph
In graph theory, an orientation of an undirected graph is an assignment of a direction to each edge, turning the initial graph into a directed graph. A
Orientation_(graph_theory)
Natural number
{\displaystyle {\mathcal {S}}} -perfect number. A Golomb ruler of length 6 is a "perfect ruler". The six exponentials theorem guarantees that under certain
6
Solution concept of a non-cooperative game
Kakutani fixed-point theorem in his 1950 paper to prove existence of equilibria. His 1951 paper used the simpler Brouwer fixed-point theorem for the same purpose
Nash_equilibrium
inequality is used, for example, in graph theory to obtain upper bounds for the number of perfect matchings in a bipartite graph. The permanent of a square binary
Bregman–Minc_inequality
Graph with a median for each three vertices
In graph theory, a division of mathematics, a median graph is an undirected graph in which every three vertices a {\displaystyle a} , b {\displaystyle
Median_graph
Bipartite non-Hamiltonian polyhedral graph
In graph theory, a branch of mathematics, the Herschel graph is a bipartite undirected graph with 11 vertices and 18 edges. It is a polyhedral graph (the
Herschel_graph
Concept in graph theory
In graph theory, a strongly regular graph (SRG) is a regular graph G = (V, E) with v vertices and degree k such that for some given integers λ , μ ≥ 0
Strongly_regular_graph
Partition of graph into sequence of paths
rank of the given graph. Robbins introduced the ear decomposition of 2-edge-connected graphs as a tool for proving the Robbins' theorem, that these are
Ear_decomposition
Upper bound on a graph's Shannon capacity
graphs for which they are equal, including perfect graphs. Let G = ( V , E ) {\displaystyle G=(V,E)} be a graph on n {\displaystyle n} vertices. An ordered
Lovász_number
Unsolved problem in computational complexity theory
2O(√n log n) for graphs with n vertices and relies on the classification of finite simple groups. Without this classification theorem, a slightly weaker
Graph_isomorphism_problem
Polyhedral graph with 26 vertices and 39 edges
In the mathematical field of graph theory, the 26-fullerene graph is a polyhedral graph with V = 26 vertices and E = 39 edges. Its planar embedding has
26-fullerene_graph
Topic in computer science
(for some graph H), k-colorability, and planarity. All hereditary properties are testable. Theorem (Alon & Shapira 2008). Every hereditary graph property
Property_testing
One of two types of graph
building blocks of line perfect graphs. The term "book-graph" has been employed for other uses. Barioli used it to mean a graph composed of a number of
Book_(graph_theory)
Measurement of graph sparsity
In graph theory, a k-degenerate graph is an undirected graph in which every subgraph has at least one vertex of degree at most k {\displaystyle k} . That
Degeneracy_(graph_theory)
French mathematician (1947-1997)
coloring and nowhere-zero flows, such as the Four color theorem, the Strong perfect graph theorem, and Tutte's 3-flow, 4-flow, and 5-flow conjectures. Gradually
François_Jaeger
undirected graph (if it exists) can be found in polynomial time by transforming the problem into a problem of finding a perfect matching in a larger graph. If
Vertex_cycle_cover
British-Canadian codebreaker and mathematician (1917–2002)
Even though Tutte's contributions to graph theory have been influential to modern graph theory and many of his theorems have been used to keep making advances
W._T._Tutte
Family of graphs based on the Fibonacci sequence
resonance graph or (Z-transformation graph) of G is a graph whose vertices describe perfect matchings of G and whose edges connect pairs of perfect matchings
Fibonacci_cube
Theorem that deals with the decompositions of complete hypergraphs
{2}{n}}=n-1} colors so that the edges of each color form a perfect matching. Baranyai's theorem says that we can do this whenever n {\displaystyle n} is
Baranyai's_theorem
Process forming a path from many random steps
abelian covering graphs over finite graphs). Actually it is possible to establish the central limit theorem and large deviation theorem in this setting
Random_walk
Assignment of colors to edges of a graph
of a given graph is called the chromatic index of the graph. By Vizing's theorem, the number of colors needed to edge color a simple graph is either its
Edge_coloring
Tool for working with matrices
cardinality matching. Kőnig's theorem is equivalent to the following: The positivity graph of any bistochastic matrix admits a perfect matching. A matrix is called
Birkhoff_algorithm
graphs are in general not chordal as the induced cycle of length 4 shows. Chordal bipartite graphs have various characterizations in terms of perfect
Chordal_bipartite_graph
Optimization technique
solving a maximum flow problem in a graph (and thus, by the max-flow min-cut theorem, define a minimal cut of the graph). Under most formulations of such
Graph cuts in computer vision and artificial intelligence
Graph_cuts_in_computer_vision_and_artificial_intelligence
PERFECT GRAPH-THEOREM
PERFECT GRAPH-THEOREM
Girl/Female
Spanish
Perfect.
Boy/Male
Hebrew, Hindu, Indian, Marathi
Grape Cluster
Boy/Male
Arabic, Modern
Grape
Girl/Female
Indian
Grape vine
Boy/Male
Muslim
Grape
Surname or Lastname
English (Bristol)
English (Bristol) : variant of Parrott 1.
Boy/Male
Tamil
Perfect
Surname or Lastname
French
French : from a pet form of the personal name P(i)erre, French form of Peter.English (Bristol) : variant of Parrott
Girl/Female
Indian
Grape like
Girl/Female
Arabic, Assamese, Hindu, Indian, Kannada, Malayalam, Marathi, Muslim, Telugu
Grape
Girl/Female
Muslim
Grape vine
Girl/Female
Hindu
Perfect
Boy/Male
Indian
Grape
Surname or Lastname
English
English : variant of Parfitt.
Girl/Female
Tamil
Surasti | ஸà¯à®°à®¸à¯à®¤à¯€
Perfect
Surasti | ஸà¯à®°à®¸à¯à®¤à¯€
Boy/Male
African, Arabic
Grape Vines
Boy/Male
Tamil
Perfect
Girl/Female
Tamil
Perfect
Girl/Female
Tamil
Nicika | நீஸீகா  Â
Perfect
Nicika | நீஸீகா  Â
Girl/Female
Muslim
Grape like
PERFECT GRAPH-THEOREM
PERFECT GRAPH-THEOREM
Boy/Male
African, German, Hindu, Indian
Order of Ram
Girl/Female
Polish
God's gracious gift.
Girl/Female
Indian
Force to move forward, Force
Boy/Male
Assamese, Bengali, Gujarati, Hindu, Indian, Kannada, Malayalam, Marathi, Oriya, Punjabi, Sanskrit, Sikh, Telugu, Traditional
Worshipped; Adoration; One who is Absorbed in God's Love
Girl/Female
German, Hebrew
Devoted to God
Girl/Female
Australian
Goddess
Boy/Male
Hindu
Sight
Boy/Male
Australian, French, German, Italian, Latin, Portuguese
War Contest; Of Mars; The Roman Fertility God Mars for whom March was Named; Warlike
Male
Egyptian
, a grandson of Tetet.
Boy/Male
Biblical
Who rules a crowd.
PERFECT GRAPH-THEOREM
PERFECT GRAPH-THEOREM
PERFECT GRAPH-THEOREM
PERFECT GRAPH-THEOREM
PERFECT GRAPH-THEOREM
a.
Well informed; certain; sure.
a.
Imperfect.
n.
One who, or that which, makes perfect.
n.
The perfect tense, or a form in that tense.
imp. & p. p.
of Perfect
v. t.
To make imperfect.
a.
Not perfect; not complete in all its parts; wanting a part; deective; deficient.
n.
The imperfect tense; or the form of a verb denoting the imperfect tense.
n.
A Roman officer who controlled or superintended a particular command, charge, department, etc.; as, the prefect of the aqueducts; the prefect of a camp, of a fleet, of the city guard, of provisions; the pretorian prefect, who was commander of the troops guarding the emperor's person.
a.
Brought to consummation or completeness; completed; not defective nor redundant; having all the properties or qualities requisite to its nature and kind; without flaw, fault, or blemish; without error; mature; whole; pure; sound; right; correct.
n.
Failing; fault; imperfection, whether physical or moral; blemish; as, a defect in the ear or eye; a defect in timber or iron; a defect of memory or judgment.
a.
Liable to defect; imperfect.
a.
To make perfect; to finish or complete, so as to leave nothing wanting; to give to anything all that is requisite to its nature and kind.
n.
Power to produce results; efficiency; force; importance; account; as, to speak with effect.
a.
Perfect.
a.
Hermaphrodite; having both stamens and pistils; -- said of flower.
a.
Producing mathematically perfect harmony or concord; sweetly or perfectly harmonious.
n.
Execution; performance; realization; operation; as, the law goes into effect in May.
adv.
In a perfect manner or degree; in or to perfection; completely; wholly; throughly; faultlessly.