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Graph Theory

Combinatorics and Graph Theory

Graph theory is the study of graphs, mathematical structures of vertices connected by edges, used to model pairwise relations between objects. It originated with Leonhard Euler's 1736 solution to the Seven Bridges of Konigsberg problem, and has grown from a branch of combinatorics into a stand-alone field with applications throughout computer science and network science. This description is adapted from Wikipedia contributors under CC BY-SA 4.0; changes were made. https://creativecommons.org/licenses/by-sa/4.0/

Facts
Central Question
Which structural properties of a network of vertices and edges, connectivity, colorability, the existence of an optimal path or cycle, can be established from its combinatorial structure alone, independent of what the vertices and edges are taken to represent? 1
Key Debate
Whether graph theory is best understood as a subfield of combinatorics or as a stand-alone branch in its own right; its rapid post-1950s growth, driven by applications in computer science, operations research and network science, gave it a body of problems distinct enough that most mathematicians now treat it as independent. 1
Classification
Pure or Applied
Both / Interdisciplinary 1
Graph Theory
Filter Results43 entries
Connections

Associated With

Four Color Theorem, Theorems

The theorem is already In Branch topology (its long proof history is topological) but its statement, that any planar map can be colored with four colors so that no two bordering regions match, is the textbook graph-coloring result once the map is translated to its dual graph; graph-theory's own key-debate names exactly this kind of colorability question as central to the branch. A second branch tie, not a replacement for the topology one.

Leonhard Euler, Mathematicians

Founding problem: the Seven Bridges of Konigsberg, 1736.

Source Graph Theory (Wikipedia)

Includes

Source Approximate max-flow min-cut theorem (Wikipedia)
Source Berge's Theorem (Wikipedia)
BEST Theorem, Theorems
Source BEST theorem (Wikipedia)
Source Betweenness Centrality (Wikipedia)
Source Hamiltonian path (Wikipedia)
Source Chromatic Polynomial (Wikipedia)
Source Wikipedia: Complete graph
Source De Bruijn graph (Wikipedia)
Source Degree Matrix (Wikipedia)
Source Dénes Kőnig (Wikipedia)
Source Hamiltonian path (Wikipedia)
Source Erdős-Faber-Lovász conjecture (Wikipedia)
Source Wikipedia: Erdos-Gyarfas Conjecture
Source Erdos-Hajnal Conjecture (Wikipedia)
Source Erdos-Posa theorem (Wikipedia)
Source Erdos-Stone theorem, Wikipedia
Source Euler Tour Technique (Wikipedia)
Source Fary's theorem (Wikipedia)
Source Five color theorem (Wikipedia)
Source Friendship graph (Wikipedia)
Source Goldberg-Seymour Conjecture (Wikipedia)
Source Graph (Discrete Mathematics) (Wikipedia)
Source Graph Automorphism (Wikipedia)
Source Graph Bandwidth (Wikipedia)
Source Graph Isomorphism (Wikipedia)
Source Graph structure theorem (Wikipedia)
Source Grinberg's theorem (Wikipedia)
Source Grötzsch Graph (Wikipedia)
Source Grotzsch's theorem (Wikipedia)
Source Equitable coloring (Wikipedia)
Source Handshaking Lemma (Wikipedia)
Source Heawood Graph (Wikipedia)
Source Hedetniemi's Conjecture (Wikipedia)
Source Heinrich Tietze (Wikipedia)
Hypergraph, Concepts
Source Julius Petersen (Wikipedia)
Source Kahn-Kalai Conjecture (Wikipedia)
Source Klaus Wagner (Wikipedia)
Source Kneser graph (Wikipedia)
Source Kotzig's theorem, Wikipedia
Source List edge-coloring (Wikipedia)
Source Lovász conjecture (Wikipedia)
Source Lovász Number (Wikipedia)
Source Turan's Theorem (Wikipedia)
Source Moser Spindle (Wikipedia)
Source Graph Coloring (Wikipedia)
Source Nash-Williams theorem (Wikipedia)
Source Perfect graph theorem (Wikipedia)
Source Petersen Graph (Wikipedia)
Source Petersen's theorem (Wikipedia)
Source Planar separator theorem (Wikipedia)
Source Reconstruction Conjecture (Wikipedia)
Source Graceful Labeling (Wikipedia)
Source Heawood conjecture (Wikipedia)
Source Ryser's Conjecture (Wikipedia)
Source Schnyder's theorem (Wikipedia)
Source Sidorenko's Conjecture (Wikipedia)
Source Strong perfect graph theorem (Wikipedia)
Source Szemeredi regularity lemma, Wikipedia
Source Total Coloring (Wikipedia)
Source Treewidth (Wikipedia)
Source Turan's Theorem (Wikipedia)
Source Tuza's Conjecture (Wikipedia)
Source Vizing's Conjecture (Wikipedia)
Source W. T. Tutte (Wikipedia)
Source Wagner's theorem - Wikipedia
Source Wikipedia: Wheel graph
Sources
1. Graph Theory (Wikipedia)
Wikipedia
  • Lead and history sections
    the study of graphs, which are mathematical structures used to model pairwise relations between objects
  • Overview of research areas
    Is it true that any map drawn in the plane may have its regions colored with four colors
  • Overview, on the field's standing
    a stand-alone field due to its great growth and distinct from other fields, having its own kind of problems
  • Lead section, pure or applied classification
    In mathematics and computer science, graph theory is the study of graphs, which are mathematical structures used to model pairwise relations between objects.
  • Associated With: Leonhard Euler, History section
    is regarded as the first paper in the history of graph theory
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Petersen Graph (Wikipedia)
Wikimedia FoundationIncludes: Petersen GraphView the Source
Reconstruction Conjecture (Wikipedia)
Wikimedia FoundationIncludes: Reconstruction Conjecture, Lead sentence
Quote, Includes: Reconstruction Conjecture, Lead sentence
In graph theory, informally, the reconstruction conjecture says that graphs are determined uniquely by their subgraphs.
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List edge-coloring (Wikipedia)
Includes: List Coloring Conjecture, Lead sentence
Quote, Includes: List Coloring Conjecture, Lead sentence
In graph theory, list edge-coloring is a type of graph coloring that combines list coloring and edge coloring.
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Graceful Labeling (Wikipedia)
Includes: Ringel-Kotzig Conjecture, Lead sentence
Quote, Includes: Ringel-Kotzig Conjecture, Lead sentence
In graph theory, a graceful labeling of a graph with m edges is a labeling of its vertices with some subset of the integers from 0
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Total Coloring (Wikipedia)
Includes: Total Coloring Conjecture, Lead sentence
Quote, Includes: Total Coloring Conjecture, Lead sentence
In graph theory, total coloring is a type of graph coloring on the vertices and edges of a graph.
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Wikipedia: Complete graph
Includes: Complete Graph, Lead sentence
Quote, Includes: Complete Graph, Lead sentence
In the mathematical field of graph theory, a complete graph is a simple undirected graph in which every pair of distinct vertices
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Wikipedia: Wheel graph
Includes: Wheel Graph, Lead sentence
Quote, Includes: Wheel Graph, Lead sentence
In graph theory, a wheel graph is a graph formed by connecting a single universal vertex to all vertices of a cycle.
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De Bruijn graph (Wikipedia)
Includes: De Bruijn Graph, Lead sentence
Quote, Includes: De Bruijn Graph, Lead sentence
In graph theory, an n-dimensional De Bruijn graph of m symbols is a directed graph representing overlaps between sequences of symb
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Petersen's theorem (Wikipedia)
Includes: Petersen's Theorem, Lead sentence
Quote, Includes: Petersen's Theorem, Lead sentence
In the mathematical discipline of graph theory, Petersen's theorem, named after Julius Petersen, is one of the earliest results in
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Hamiltonian path (Wikipedia)
  • Includes: Bondy-Chvatal Theorem, Lead sentence
    In the mathematical field of graph theory, a Hamiltonian path (or traceable path) is a path in an undirected or directed graph tha
  • Includes: Dirac's Theorem, Lead sentence
    In the mathematical field of graph theory, a Hamiltonian path (or traceable path) is a path in an undirected or directed graph tha
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BEST theorem (Wikipedia)
Includes: BEST Theorem, Lead sentence
Quote, Includes: BEST Theorem, Lead sentence
In graph theory, a part of discrete mathematics, the BEST theorem gives a product formula for the number of Eulerian circuits in d
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Five color theorem (Wikipedia)
Includes: Five Color Theorem, Lead sentence
Quote, Includes: Five Color Theorem, Lead sentence
The five color theorem is a result from graph theory that given a plane separated into regions, such as a political map of the cou
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Graph structure theorem (Wikipedia)
Includes: Graph Structure Theorem, Lead sentence
Quote, Includes: Graph Structure Theorem, Lead sentence
tructure theorem is a major result in the area of graph theory.
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Grinberg's theorem (Wikipedia)
Includes: Grinberg's Theorem, Lead sentence
Quote, Includes: Grinberg's Theorem, Lead sentence
In graph theory, Grinberg's theorem is a necessary condition for a planar graph to contain a Hamiltonian cycle, based on the lengt
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Equitable coloring (Wikipedia)
Includes: Hajnal-Szemeredi Theorem, Lead sentence
Quote, Includes: Hajnal-Szemeredi Theorem, Lead sentence
In graph theory, an area of mathematics, an equitable coloring is an assignment of colors to the vertices of an undirected graph,
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Graph Coloring (Wikipedia)
Wikimedia FoundationIncludes: Mycielski's Theorem, Lead sentence
Quote, Includes: Mycielski's Theorem, Lead sentence
In graph theory, graph coloring is a methodic assignment of labels traditionally called "colors" to elements of a graph.
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Planar separator theorem (Wikipedia)
Includes: Planar Separator Theorem, Lead sentence
Quote, Includes: Planar Separator Theorem, Lead sentence
In graph theory, the planar separator theorem is a form of isoperimetric inequality for planar graphs, that states that any planar
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Graph Isomorphism (Wikipedia)
Includes: Graph Isomorphism, Lead sentence
Quote, Includes: Graph Isomorphism, Lead sentence
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 ) such tha
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Heawood Graph (Wikipedia)
Includes: Heawood Graph, Lead sentence
Quote, Includes: Heawood Graph, Lead sentence
In the mathematical field of graph theory, the Heawood graph is an undirected graph with 14 vertices and 21 edges, named after Per
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Graph Automorphism (Wikipedia)
Includes: Graph Automorphism, Lead sentence
Quote, Includes: Graph Automorphism, Lead sentence
In the mathematical field of graph theory, an automorphism of a graph is a form of symmetry in which the graph is mapped onto itse
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Graph Bandwidth (Wikipedia)
Includes: Graph Bandwidth, Lead sentence
Quote, Includes: Graph Bandwidth, Lead sentence
In graph theory, the graph bandwidth problem may be visualized as placing the vertices of a given graph at distinct integer positi
View the Source
Moser Spindle (Wikipedia)
Includes: Moser Spindle, Lead sentence
Quote, Includes: Moser Spindle, Lead sentence
In graph theory, a branch of mathematics, the Moser spindle (also called the Mosers' spindle or Moser graph) is an undirected grap
View the Source
Chromatic Polynomial (Wikipedia)
Includes: Chromatic Polynomial, Lead sentence
Quote, Includes: Chromatic Polynomial, Lead sentence
nomial is a graph polynomial studied in algebraic graph theory, a branch of mathematics.
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Degree Matrix (Wikipedia)
Includes: Degree Matrix, Lead sentence
Quote, Includes: Degree Matrix, Lead sentence
In the mathematical field of algebraic graph theory, the degree matrix of an undirected graph is a diagonal matrix which contains
View the Source
Goldberg-Seymour Conjecture (Wikipedia)
Includes: Goldberg-Seymour Conjecture, Lead sentenceView the Source
Erdos-Posa theorem (Wikipedia)
Includes: Erdos-Posa Theorem, Lead sentenceView the Source
Vizing's Conjecture (Wikipedia)
Wikimedia FoundationIncludes: Vizing's Conjecture, Lead sentence
Quote, Includes: Vizing's Conjecture, Lead sentence
In graph theory, Vizing's conjecture concerns a relation between the domination number and the cartesian product of graphs.
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Tuza's Conjecture (Wikipedia)
Wikimedia FoundationIncludes: Tuza's Conjecture, Lead sentence
Quote, Includes: Tuza's Conjecture, Lead sentence
Tuza's conjecture is an unsolved problem in graph theory, a branch of mathematics, concerning triangles in undirected graphs.
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Lovász conjecture (Wikipedia)
Includes: Lovasz Conjecture, Lead sentenceView the Source
Wikipedia: Erdos-Gyarfas Conjecture
WikipediaIncludes: Erdos-Gyarfas Conjecture, Lead sentenceView the Source
Erdős-Faber-Lovász conjecture (Wikipedia)
Includes: Erdos-Faber-Lovasz Conjecture, Lead sentenceView the Source
Hedetniemi's Conjecture (Wikipedia)
Wikimedia FoundationIncludes: Hedetniemi's Conjecture, Lead sentence
Quote, Includes: Hedetniemi's Conjecture, Lead sentence
In graph theory, Hedetniemi's conjecture, formulated by Stephen T.
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Kahn-Kalai Conjecture (Wikipedia)
Includes: Kahn-Kalai Conjecture, Lead sentence
Quote, Includes: Kahn-Kalai Conjecture, Lead sentence
rk-Pham Theorem, was a conjecture in the field of graph theory and statistical mechanics, proposed by Jeff Kahn and Gil Kalai in 2
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Ryser's Conjecture (Wikipedia)
Includes: Ryser's Conjecture, Lead sentence
Quote, Includes: Ryser's Conjecture, Lead sentence
In graph theory, Ryser's conjecture is a conjecture relating the maximum matching size and the minimum transversal size in hypergr
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Sidorenko's Conjecture (Wikipedia)
Includes: Sidorenko's Conjecture, Lead sentence
Quote, Includes: Sidorenko's Conjecture, Lead sentence
re is a major conjecture in the field of extremal graph theory, posed by Alexander Sidorenko in 1986.
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Erdos-Hajnal Conjecture (Wikipedia)
Includes: Erdos-Hajnal Conjecture, Lead sentenceView the Source
Turan's Theorem (Wikipedia)
Wikimedia Foundation
  • Includes: Turan Graph, Lead sentence
  • Includes: Mantel's Theorem, Lead sentence
View the Source
Kneser graph (Wikipedia)
Includes: Kneser Graph, Lead sentence
Quote, Includes: Kneser Graph, Lead sentence
In graph theory, the Kneser graph K(n, k) (alternatively KGn,k) is the graph whose vertices correspond to the k-element subsets of
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Handshaking Lemma (Wikipedia)
Wikimedia FoundationIncludes: Handshaking Lemma, Lead sentence
Quote, Includes: Handshaking Lemma, Lead sentence
In graph theory, the handshaking lemma is the statement that, in every finite undirected graph, the number of vertices that touch
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Szemeredi regularity lemma, Wikipedia
Includes: Szemeredi Regularity Lemma, Lead sentenceView the Source
Friendship graph (Wikipedia)
Includes: Friendship Theorem, Lead sentence
Quote, Includes: Friendship Theorem, Lead sentence
In the mathematical field of graph theory, the friendship graph (or Dutch windmill graph or n-fan) Fn is a planar, undirected grap
View the Source
Wagner's theorem - Wikipedia
Includes: Wagner's Theorem, Lead sentence
Quote, Includes: Wagner's Theorem, Lead sentence
In graph theory, Wagner's theorem is a mathematical forbidden graph characterization of planar graphs, named after Klaus Wagner, s
View the Source
Berge's Theorem (Wikipedia)
Wikimedia FoundationIncludes: Berge's Theorem, Lead sentence
Quote, Includes: Berge's Theorem, Lead sentence
In graph theory, Berge's theorem states that a matching M in a graph G is maximum (contains the largest possible number of edges)
View the Source
Erdos-Stone theorem, Wikipedia
Includes: Erdos-Stone Theorem, Lead sentenceView the Source
Nash-Williams theorem (Wikipedia)
Includes: Nash-Williams Formula, Lead sentence
Quote, Includes: Nash-Williams Formula, Lead sentence
In graph theory, the Nash-Williams theorem is a tree-packing theorem that describes how many edge-disjoint spanning trees (and mor
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Fary's theorem (Wikipedia)
Includes: Fary's Theorem, Lead sentenceView the Source
Kotzig's theorem, Wikipedia
Includes: Kotzig's Theorem, Lead sentence
Quote, Includes: Kotzig's Theorem, Lead sentence
In graph theory and polyhedral combinatorics, areas of mathematics, Kotzig's theorem is the statement that every polyhedral graph
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Approximate max-flow min-cut theorem (Wikipedia)
Wikimedia FoundationIncludes: Approximate Max-Flow Min-Cut Theorem, Lead sentence
Quote, Includes: Approximate Max-Flow Min-Cut Theorem, Lead sentence
In graph theory, approximate max-flow min-cut theorems concern the relationship between the maximum flow rate (max-flow) and the m
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Grotzsch's theorem (Wikipedia)
Includes: Grotzsch's Theorem, Lead sentenceView the Source
Perfect graph theorem (Wikipedia)
Includes: Perfect Graph Theorem, Lead sentenceView the Source
Heawood conjecture (Wikipedia)
Includes: Ringel-Youngs Theorem, Lead sentenceView the Source
Schnyder's theorem (Wikipedia)
Includes: Schnyder's Theorem, Lead sentence
Quote, Includes: Schnyder's Theorem, Lead sentence
In graph theory, Schnyder's theorem is a characterization of planar graphs in terms of the order dimension of their incidence pose
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Strong perfect graph theorem (Wikipedia)
Includes: Strong Perfect Graph Theorem, Lead sentence
Quote, Includes: Strong Perfect Graph Theorem, Lead sentence
In graph theory, the strong perfect graph theorem is a forbidden graph characterization of the perfect graphs as being exactly the
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Grötzsch Graph (Wikipedia)
Includes: Grötzsch Graph, Lead sentenceView the Source
Treewidth (Wikipedia)
Includes: Treewidth, Lead sentence
Quote, Includes: Treewidth, Lead sentence
In graph theory, the treewidth of an undirected graph is an integer number which specifies, informally, how far the graph is from
View the Source
Lovász Number (Wikipedia)
Includes: Lovász Number, Lead sentenceView the Source
Euler Tour Technique (Wikipedia)
Includes: Euler Tour Technique, Lead sentence
Quote, Includes: Euler Tour Technique, Lead sentence
(ETT), named after Leonhard Euler, is a method in graph theory for representing trees.
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Betweenness Centrality (Wikipedia)
Includes: Betweenness Centrality, Lead sentence
Quote, Includes: Betweenness Centrality, Lead sentence
In graph theory, betweenness centrality is a measure of centrality in a graph based on shortest paths.
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W. T. Tutte (Wikipedia)
Includes: W. T. Tutte, Lead paragraph [in-branch]
Quote, Includes: W. T. Tutte, Lead paragraph [in-branch]
graph theory
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Dénes Kőnig (Wikipedia)
Includes: Denes Konig, Lead paragraph
Quote, Includes: Denes Konig, Lead paragraph
graph theory
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Julius Petersen (Wikipedia)
Includes: Julius Petersen, Lead paragraph [in-branch]
Quote, Includes: Julius Petersen, Lead paragraph [in-branch]
graph theory
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Heinrich Tietze (Wikipedia)
Includes: Heinrich Tietze, Lead paragraph [in-branch 3]
Quote, Includes: Heinrich Tietze, Lead paragraph [in-branch 3]
Tietze's graph
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Klaus Wagner (Wikipedia)
Includes: Klaus Wagner, Lead paragraph [in-branch]
Quote, Includes: Klaus Wagner, Lead paragraph [in-branch]
graph theory
View the Source
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