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Assignment of colors to edges of a graph
graph theory, a proper edge coloring of a graph is an assignment of "colors" to the edges of the graph so that no two incident edges have the same color
Edge_coloring
Methodic assignment of colors to elements of a graph
vertex coloring. Similarly, an edge coloring assigns a color to each edge so that no two adjacent edges are of the same color, and a face coloring of a
Graph_coloring
Graph edge coloring with a limited number of allowed colors
theory, list edge-coloring is a type of graph coloring that combines list coloring and edge coloring. An instance of a list edge-coloring problem consists
List_edge-coloring
Algorithm in graph theory
& Gries edge-coloring algorithm is a polynomial-time algorithm in graph theory that finds an edge coloring of any simple graph. The coloring produced
Misra & Gries edge-coloring algorithm
Misra_&_Gries_edge-coloring_algorithm
Coloring in which edges are labeled by integers
In graph theory, interval edge coloring is a type of edge coloring in which edges are labeled by the integers in some interval, every integer in the interval
Interval_edge_coloring
Cubic graph with 10 vertices and 15 edges
constructed it to be the smallest bridgeless cubic graph with no three-edge-coloring. Although the graph is generally credited to Petersen, it had in fact
Petersen_graph
Three-vertex regular multigraph
are a special type of triangle graphs, which are used in the field of edge coloring in particular. A Shannon multigraph is multigraph with 3 vertices for
Shannon_multigraph
2003.05.001, MR 2035386 Fouquet, J.-L.; Jolivet, J.-L. (1983), "Strong edge-colorings of graphs and applications to multi-k-gons", Ars Combinatoria, 16 (A):
Induced_matching
Generalization of graph coloring to the hypergraph
Conflict-free 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
Conflict-free_coloring
Undirected cubic graph with 12 vertices and 18 edges
remains non-Hamiltonian. Edge coloring Tietze's graph requires four colors; that is, its chromatic index is 4. Equivalently, the edges of Tietze's graph can
Tietze's_graph
Graph coloring of both the edges and vertices
theory, total coloring is a type of graph coloring on the vertices and edges of a graph. When used without any qualification, a total coloring is always assumed
Total_coloring
Minimum monochromatic-triangle theorem in graph theory
minimum number of monochromatic triangles in a two-coloring of the edges of a complete graph. If the edges of K n {\displaystyle K_{n}} are colored red or
Goodman's_theorem
Family of symmetric graphs which generalize the Petersen graph
each game represents an edge of O 6 {\displaystyle O_{6}} , each weekday is represented by a color, and a 6-color edge coloring of O 6 {\displaystyle O_{6}}
Odd_graph
Theorem in combinatorics
in the language of list coloring. For a graph G {\displaystyle G} , a list assignment L {\displaystyle L} attaches to every edge e {\displaystyle e} a set
Dinitz_theorem
Graph divided into two independent sets
a coloring of the graph with two colors: if one colors all nodes in U {\displaystyle U} blue, and all nodes in V {\displaystyle V} red, each edge has
Bipartite_graph
of coloring have been studied, including edge coloring (coloring edges so that no two edges with the same endpoint share a color), list coloring (proper
Glossary_of_graph_theory
edge. The star edge coloring of a graph G is a proper edge coloring without bichromatic paths and cycles of length four, similar to the star coloring
Corona_product
Graph coloring where each vertex has a list of allowed colors
In graph theory, a branch of mathematics, list coloring is a type of graph coloring where each vertex can be restricted to a list of allowed colors. It
List_coloring
Family of cubic graphs formed from regular and star polygons
to have only one 3-edge-coloring. A 4-edge-coloring of the Petersen graph or G ( 5 , 2 ) {\displaystyle G(5,2)} A 3-edge-coloring of the Dürer graph or
Generalized_Petersen_graph
On coloring the edges of graphs
proper (Δ+1)-edge-coloring of G from c. The other way around, if a proper (Δ+1)-edge-coloring exists, then we can delete xy, restrict the coloring and (1)
Vizing's_theorem
Path on an edge-colored graph over which no color repeats
{\displaystyle 2\leq k\leq n} . An edge coloring of G {\displaystyle G} is called a k {\displaystyle k} -rainbow coloring if for every set S {\displaystyle
Rainbow_coloring
3-regular graph with no 3-edge-coloring
requirements for graph edge coloring. Additionally, a cycle consisting of two vertices connected by two edges can always be replaced by a single edge connecting their
Snark_(graph_theory)
Bipartite graph where each node of 1st set is linked to all nodes of 2nd set
matching of size min{m,n}. A complete bipartite graph Kn,n has a proper n-edge-coloring corresponding to a Latin square. Every complete bipartite graph is a
Complete_bipartite_graph
Class of mathematical games
the vertex coloring game on a graph G with k colors. Does she have one for k+1 colors? More unsolved problems in mathematics The graph coloring game is a
Graph_coloring_game
Area of discrete mathematics
possible Cycle double cover, a collection of cycles covering each edge exactly twice Edge coloring, a decomposition into as few matchings as possible Graph factorization
Graph_theory
Partition of a graph into spanning subgraphs
perfect matching, and a 1-factorization of a k-regular graph is a proper edge coloring with k colors. A 2-factor is a collection of disjoint cycles that spans
Graph_factorization
Graph coloring problem on paths in a network
coloring is a type of graph coloring where colors (or wavelengths) are assigned to a set of paths in a graph such that any two paths sharing an edge receive
Path_coloring
Statement in mathematical combinatorics
λ)-graph G with small λ and edge density 1⁄2 contains an induced monochromatic copy of every graph on k vertices in any edge coloring in two colors. Currently
Ramsey's_theorem
Undirected unit-distance graph requiring four colors
seven vertices and eleven edges. It can be drawn as a unit distance graph, and it requires four colors in any graph coloring. Its existence can be used
Moser_spindle
Non-crossing graph with vertices on outer face
maximum degree except when the graph forms a cycle of odd length. An edge coloring with an optimal number of colors can be found in linear time based on
Outerplanar_graph
triangle-free edge-coloring when there are exactly two colors available. If there exists a two-color triangle-free edge coloring, then the edges of each color
Monochromatic_triangle
Influence of local substructure of a graph on global properties
graph G {\displaystyle G} is the minimum number of colors in a proper edge-coloring of a graph, and Vizing's theorem states that the chromatic index of
Extremal_graph_theory
One-by-one assignment of colors to graph vertices
the study of graph coloring problems in mathematics and computer science, a greedy coloring or sequential coloring is a coloring of the vertices of a
Greedy_coloring
Disproven graph theory
problem of finding 3-edge-colorings of bridgeless cubic planar graphs. In a Hamiltonian cubic planar graph, such an edge coloring is easy to find: use
Tait's_conjecture
Goldberg–Seymour conjecture Graph coloring game Graph two-coloring Harmonious coloring Incidence coloring List coloring List edge-coloring Perfect graph Ramsey's
List_of_graph_theory_topics
Graph coloring where graph elements are assigned sets of colors
In a traditional graph coloring, each vertex in a graph is assigned some color, and adjacent vertices — those connected by edges — must be assigned different
Fractional_coloring
Edge-colored graph matching where all edges have distinct colors
edge-coloring is called proper if each edge has a single color, and each two edges of the same color have no vertex in common. A proper edge-coloring
Rainbow_matching
Graph with all vertices of degree 3
3-edge-coloring is known as a Tait coloring, and forms a partition of the edges of the graph into three perfect matchings. By Kőnig's line coloring theorem
Cubic_graph
Measurement of graph sparsity
orientation can be formed by orienting each edge towards the earlier of its two endpoints in a coloring number ordering. In the other direction, if an
Degeneracy_(graph_theory)
Book containing art to be colored in by the user
artistic media. Traditional coloring books and coloring pages are printed on paper or card. Some coloring books have perforated edges so their pages can be
Coloring_book
Vertex coloring where every color pairing appears at least once
In graph theory, a complete coloring is a (proper) vertex coloring in which every pair of colors appears on at least one pair of adjacent vertices. Equivalently
Complete_coloring
Semiregular tiling of the Euclidean plane
one uniform coloring in a rhombitrihexagonal tiling. (Naming the colors by indices around a vertex (3.4.6.4): 1232.) With edge-colorings there is a half
Rhombitrihexagonal_tiling
Unproven generalization of the four-color theorem
incident edges; one can 3-color any such graph by removing one such vertex, coloring the remaining graph recursively, and then adding back and coloring the
Hadwiger conjecture (graph theory)
Hadwiger_conjecture_(graph_theory)
Graph coloring related to treedepth
theory, a centered coloring is a type of graph coloring related to treedepth. The minimum number of colors in a centered coloring of a graph equals the
Centered_coloring
Graph with only one possible coloring
colorable subgraph. A uniquely edge-colorable graph is a k-edge-chromatic graph that has only one possible (proper) k-edge-coloring up to permutation of the
Uniquely_colorable_graph
Probability saying
Threshold for Random Graphs with a Monochromatic Triangle in Every Edge Coloring". Memoirs of the American Mathematical Society. 179 (845). AMS Bookstore:
Almost_surely
Graph coloring with one edge per color pair
instance, a path with three edges has a complete 3-coloring. Exact colorings are closely related to harmonious colorings (colorings in which each pair of colors
Exact_coloring
illustrates an amalgamation of K 5 {\displaystyle K_{5}} . The invariance of edge coloring and Hamiltonian Decomposition can be seen clearly. The function ϕ {\displaystyle
Graph_amalgamation
Concept in graph theory
flow that is nowhere zero. It is intimately connected (by duality) to coloring planar graphs. Let G = (V,E) be a digraph and let M be an abelian group
Nowhere-zero_flow
On graph coloring and neighborhood size
degree of a graph also appears in upper bounds for other types of coloring; for edge coloring, the result that the chromatic index is at most Δ + 1 is Vizing's
Brooks'_theorem
polynomial-time edge coloring algorithm achieving the conjectured bound. Petersen graph#Coloring Fractional coloring Graph coloring "Problems in Graph
Goldberg–Seymour_conjecture
Graph with at most one crossing per edge
complicated. Ringel's motivation was in trying to solve a variation of total coloring for planar graphs, in which one simultaneously colors the vertices and
1-planar_graph
Property of graphs that depends only on abstract structure
for the vertices in a proper coloring Chromatic index, the smallest number of colors for the edges in a proper edge coloring Choosability (or list chromatic
Graph_property
Graph describing a topological embedding
cubic graph H {\displaystyle H} together with a 3-edge-coloring of H {\displaystyle H} . Each edge e {\displaystyle e} of G {\displaystyle G} is expanded
Graph-encoded_map
Matrix with symbols that each occur once per row and column
Latin squares may also be described as the optimal colorings of rook's graphs, or as optimal edge colorings of complete bipartite graphs. An example of a 3
Latin_rectangle
Cycles in a graph that cover each edge twice
the graph into three perfect matchings (that is, the graph has no 3-edge coloring, and by Vizing's theorem has chromatic index 4). It turns out that snarks
Cycle_double_cover
Graph of chess rook moves
164.51, JSTOR 20159988, S2CID 119151552. For the equivalence between edge-coloring complete bipartite graphs and Latin squares, see e.g. LeSaulnier, Timothy
Rook's_graph
Assignment of labels to elements of a graph
harmonious. A graph coloring is a subclass of graph labelings. Vertex colorings assign different labels to adjacent vertices, while edge colorings assign different
Graph_labeling
Concept in graph theory
an incidence coloring of G {\displaystyle G} . It is equivalent to a strong edge coloring of the graph obtained by subdivising each edge of G {\displaystyle
Incidence_(graph)
Special labeling in graph theory
theory, the act of coloring generally implies the assignment of labels to vertices, edges or faces in a graph. The incidence coloring is a special graph
Incidence_coloring
Set of edges without common vertices
that each edge belongs to a perfect matching if and only if its endpoints belong to the same subset Edge coloring, a partition of the edges of a graph
Matching_(graph_theory)
Pattern in mathematics and computer science
Source: Given a simple graph G = ( V , E ) {\displaystyle G=(V,E)} , a edge coloring c : E → Δ {\displaystyle c:E\rightarrow \Delta } matches pattern p {\displaystyle
Unavoidable_pattern
number. Alon et al. define a nonrepetitive coloring of a graph to be an assignment of colors to the edges of the graph, such that there does not exist
Thue_number
Algorithms and methods of plotting the Mandelbrot set on a computing device
show structures of the data (scientific visualisation) A more complex coloring method involves using a histogram which pairs each pixel with said pixel's
Plotting algorithms for the Mandelbrot set
Plotting_algorithms_for_the_Mandelbrot_set
Conjecture about coloring graphs
forms a simple hypergraph. And, the hypergraph dual of vertex coloring is edge coloring. Thus, the Erdős–Faber–Lovász conjecture is equivalent to the
Erdős–Faber–Lovász_conjecture
edge coloring of graphs with maximum degree Δ {\displaystyle \Delta } , it was first shown using the local lemma that there always exists a coloring with
Entropy_compression
Planar maps require at most four colors
four colors or fewer, so is the original graph since the same coloring is valid if edges are removed. So it suffices to prove the four color theorem for
Four_color_theorem
Theorem in graph theory
for a synchronizing coloring to exist.) The edges of this graph have been colored red and blue to create a synchronizing coloring. For example, consider
Road_coloring_theorem
Problem in quantum information science
the Hamiltonian is represented as a Sparse matrix, the distributed edge coloring algorithm can be used to decompose it into a sum of terms; which can
Hamiltonian_simulation
Theorem that deals with the decompositions of complete hypergraphs
even number of vertices has an edge coloring whose number of colors equals its degree, or equivalently that its edges may be partitioned into perfect
Baranyai's_theorem
Book
chapters concern graph coloring, the history of the four color theorem for planar graphs, its equivalence to 3-edge-coloring of planar cubic graphs,
The_Petersen_Graph
is a bridgeless cubic graph that requires four colors in any proper edge coloring. The smallest snark is the Petersen graph, already listed above. Blanuša
List_of_graphs
Square array with symbols that each occur once per row and column
is an edge (between its row and its column), and the symbols are colors. The rules of the Latin squares imply that this is a proper edge coloring. With
Latin_square
Complexity class of approximable problems
One other example of a potentially APX-intermediate problem is min edge coloring. One can also define a family of complexity classes f ( n ) {\displaystyle
APX
1979 classic textbook on computational complexity theory
(link) NP-complete: Holyer, Ian (November 1981). "The NP-Completeness of Edge-Coloring". SIAM Journal on Computing. 10 (4): 718–720. doi:10.1137/0210055. In
Computers_and_Intractability
Type of graph
vertex by a single edge and having degree three at its other vertices. This construction produces a graph that has no 3-color edge coloring, regardless of
Loupekine_snark
On coloring infinite graphs
The graph is finite when its vertices and edges form finite sets, and infinite otherwise. A graph coloring associates each vertex with a color drawn from
De Bruijn–Erdős theorem (graph theory)
De_Bruijn–Erdős_theorem_(graph_theory)
Natural number
can be reached by a knight within three moves. There are 109 uniform edge-colorings to the 11 regular and semiregular (or Archimedean) tilings. The decimal
109_(number)
Special type of graph coloring
of directed graph, with at most one directed edge between each pair of vertices. An oriented graph coloring of a graph G {\displaystyle G} can be described
Oriented_coloring
Graph with a prism as its skeleton
1-factorization is a partition of the edge set of the graph into three perfect matchings, or equivalently an edge coloring of the graph with three colors. Every
Prism_graph
Function in algebraic graph theory
G-uv} denote the graph obtained by removing the edge u v {\displaystyle uv} . Then the numbers of k-colorings of these graphs satisfy: P ( G , k ) = P ( G
Chromatic_polynomial
Orange-red condiment and food coloring derived from the seeds of the achiote tree
Annatto (/əˈnætoʊ/ or /əˈnɑːtoʊ/) is an orange-red condiment and food coloring derived from the seeds of the achiote tree (Bixa orellana), native to tropical
Annatto
Process of partitioning a rectilinear polygon
guillotine-cut (also called an edge-to-edge cut) is a straight bisecting line going from one edge of an existing polygon to the opposite edge, similarly to a paper
Guillotine_partition
on graphs. Coloring algorithm: algorithms for graph (vertex or edge) coloring (subject to constraints, e.g. proper coloring or list coloring) Hopcroft–Karp
List_of_algorithms
Subgraph with contracted edges
that any bridgeless 3-regular graph that requires four colors in an edge coloring must have the Petersen graph as a minor. Many families of graphs have
Graph_minor
graphs are class 2. That is, they require at least Δ + 1 colors in any edge coloring. A graph G, with an overfull subgraph S such that Δ ( G ) = Δ ( S )
Overfull_graph
Infinite graph containing all countable graphs
extended to edge-colored graphs; that is, graphs in which the edges have been assigned to different color classes, but without the usual edge coloring requirement
Rado_graph
graph theory, a branch of mathematics, a radio coloring of an undirected graph is a form of graph coloring in which one assigns positive integer labels
Radio_coloring
Graph with directed and undirected edges
(undirected) edges E, and a set of directed edges (or arcs) A. Consider adjacent vertices u , v ∈ V {\displaystyle u,v\in V} . A directed edge, called an
Mixed_graph
Geometric graph with unit edge lengths
There exist unit distance graphs requiring five colors in any proper coloring, and all unit distance graphs can be colored with at most seven colors
Unit_distance_graph
Theorem in graph theory and polyhedral combinatorics
O. V. (1990), "A generalization of Kotzig's theorem and prescribed edge coloring of planar graphs", Matematicheskie Zametki, 48 (6): 22–28, 160, doi:10
Kotzig's_theorem
in which all vertices have the same outdegree has a synchronizable edge coloring if and only if it is aperiodic. Jarvis, J. P.; Shier, D. R. (1996),
Aperiodic_graph
Graph coloring with an allowed number of same-color neighbors
mathematical discipline, coloring refers to an assignment of colours or labels to vertices, edges and faces of a graph. Defective coloring is a variant of proper
Defective_coloring
Hyde, John; Jensen, Melanie; Mann, Casey; Schroeder, Tyler. "Uniform edge-c-colorings of the Archimedean Tilings" (PDF). University of Washington. (Casey
List of Euclidean uniform tilings
List_of_Euclidean_uniform_tilings
Maximum number of colors in a greedy graph coloring
the independent set has at least one edge incident to it. A Grundy coloring of a t-atom can be obtained by coloring the independent set first with the smallest-numbered
Grundy_number
Graph with tight clique-coloring relation
graph, there is an edge from x {\displaystyle x} to y {\displaystyle y} whenever the two intervals have a point in common. Coloring these graphs can be
Perfect_graph
Family of sets where every disjoint subfamily has k or fewer sets
hypergraph of H has the property that its maximum degree equals its minimum edge coloring number. Bollobás, Béla (1986), Combinatorics: Set Systems, Hypergraphs
Helly_family
Theorem on triangulation graph colorings
a triangulation consisting of smaller triangles meeting edge to edge. Then a Sperner coloring of the triangulation is defined as an assignment of three
Sperner's_lemma
p/2−1 and p/2). Mirroring only works for even p. It can be proven that a coloring with the desired properties exists for all p. When mirroring is used to
Two-tree_broadcast
British mathematician
Lancaster University. Her research interests include graph theory, edge coloring, and latin squares in combinatorics, as well as geographical clustering
Amanda_Chetwynd
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