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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
Methodic assignment of colors to elements of a graph
In graph theory, graph coloring is a methodic assignment of labels traditionally called "colors" to elements of a graph. The assignment is subject to certain
Graph_coloring
Vertex coloring where no two linked nodes have the same color pairing
at most one pair of adjacent vertices. It is the opposite of the complete coloring, which instead requires every color pairing to occur at least once
Harmonious_coloring
Assignment of colors to edges of a graph
edge coloring of a graph by the colors red, blue, and green. Edge colorings are one of several different types of graph coloring. The edge-coloring problem
Edge_coloring
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
Graph coloring with equal color classes
In graph theory, an area of mathematics, an equitable coloring is an assignment of colors to the vertices of an undirected graph, in such a way that No
Equitable_coloring
include the rural postman problem. Clique cover problem Clique problem Complete coloring, a.k.a. achromatic number Cycle rank Degree-constrained spanning tree
List_of_NP-complete_problems
Graph coloring with one edge per color pair
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
Complexity class
Graph coloring problem Sudoku To the right is a diagram of some of the problems and the reductions typically used to prove their NP-completeness. In this
NP-completeness
achromatic number of a graph is the maximum number of colors in a complete coloring. acyclic 1. A graph is acyclic if it has no cycles. An undirected
Glossary_of_graph_theory
Graph coloring avoiding 2-colored paths
In the mathematical field of graph theory, a star coloring of a graph G is a (proper) vertex coloring in which every path on four vertices uses at least
Star_coloring
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
2016 mixtape by Chance the Rapper
Coloring Book is the third mixtape by American rapper Chance the Rapper. It was produced by his group The Social Experiment, Lido, and Kaytranada, among
Coloring_Book_(mixtape)
Path on an edge-colored graph over which no color repeats
{\displaystyle {\text{src}}(G)} . Clearly, each strong rainbow coloring is also a rainbow coloring, while the converse is not true in general. It is easy to
Rainbow_coloring
Graph coloring with an allowed number of same-color neighbors
and related colorings are given by Marietjie Frick. Cowen, Cowen and Woodall focused on graphs embedded on surfaces and gave a complete characterization
Defective_coloring
In graph theory, circular coloring is a kind of coloring that may be viewed as a refinement of the usual graph coloring. The circular chromatic number
Circular_coloring
Assignment of colors to graph vertices that destroys all symmetries
In graph theory, a distinguishing coloring or distinguishing labeling of a graph is an assignment of colors or labels to the vertices of the graph that
Distinguishing_coloring
On graph coloring and neighborhood size
of it in 1941. A coloring with the number of colors described by Brooks' theorem is sometimes called a Brooks coloring or a Δ-coloring. For any connected
Brooks'_theorem
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
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
Class of mathematical games
construct a coloring of a graph, following specific rules depending on the game we consider. One player tries to successfully complete the coloring of the
Graph_coloring_game
Netflix media franchise
Things: The Official Color-with-Stickers Book and Stranger Things: The Complete Coloring Book, which released on September 30, 2025. A themed cookbook, Stranger
Stranger_Things_(franchise)
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 uses
Misra & Gries edge-coloring algorithm
Misra_&_Gries_edge-coloring_algorithm
Maximum number of colors in a greedy graph coloring
the path are colored first, the greedy coloring algorithm will use three colors for the whole graph. The complete bipartite graphs are the only connected
Grundy_number
Toroidal polyhedron with 7 faces
In geometry, the Szilassi polyhedron is a nonconvex polyhedron, topologically a torus, with seven hexagonal faces. The tetrahedron and the Szilassi polyhedron
Szilassi_polyhedron
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
American rapper (born 1993)
figure in Chicago hip-hop and among independent artists. His third mixtape, Coloring Book (2016), received critical acclaim and became the first streaming-only
Chance_the_Rapper
Coloring in which edges are labeled by integers
not all graphs allow interval edge coloring. A simple family of graphs that allows interval edge coloring is complete graph of even order and a counter
Interval_edge_coloring
Type of total coloring in graph theory
In graph theory, a total coloring is a coloring on the vertices and edges of a graph such that: (1). no adjacent vertices have the same color; (2). no
Adjacent-vertex-distinguishing-total coloring
Adjacent-vertex-distinguishing-total_coloring
Set of computational problems stated by Richard Karp (1973)
per clause (equivalent to 3-SAT) Chromatic number (also called the Graph Coloring Problem) Clique cover Exact cover Hitting set Steiner tree 3-dimensional
Karp's 21 NP-complete problems
Karp's_21_NP-complete_problems
Bipartite graph where each node of 1st set is linked to all nodes of 2nd set
trees. A complete bipartite graph Km,n has a maximum matching of size min{m,n}. A complete bipartite graph Kn,n has a proper n-edge-coloring corresponding
Complete_bipartite_graph
Special labeling in graph theory
L(h, k)-coloring Harmonious coloring Star coloring Total coloring Circular coloring Path coloring Defective coloring Radio coloring Acyclic coloring
Incidence_coloring
Fantasy book series
The Complete Book of Dragons (2003) The Dragonology Handbook: A Practical Course in Dragons A Dragonology Code Writing Kit Dragonology The Coloring Book
Ology_(book_series)
Relation between graph coloring and crossings
conjectures in graph coloring theory. The conjecture states that, among all graphs requiring n {\displaystyle n} colors, the complete graph K n {\displaystyle
Albertson_conjecture
Graph coloring in which all 2-chromatic subgraphs are acyclic
In graph theory, an acyclic coloring is a (proper) vertex coloring in which every 2-chromatic subgraph is acyclic. The acyclic chromatic number A(G) of
Acyclic_coloring
Problem in graph theory
is also NP-complete for some other classes of graphs on which the usual graph coloring problem is easier. For instance it is NP-complete on the rook's
Precoloring_extension
Unsolved problem in the mathematics of graph coloring
problem in mathematics Can every two ( d + 2 ) {\displaystyle (d+2)} -colorings of a d {\displaystyle d} -degenerate graph be transformed into each other
Cereceda's_conjecture
2016 concert tour by Chance the Rapper
Magnificent Coloring World Tour was a headlining concert tour by American recording artist, Chance the Rapper, starting at the CalCoast Credit Union Open
Magnificent Coloring World Tour
Magnificent_Coloring_World_Tour
graph Acyclic coloring Chromatic polynomial Cocoloring Complete coloring Edge coloring Exact coloring Four color theorem Fractional coloring Goldberg–Seymour
List_of_graph_theory_topics
Planar maps require at most four colors
software. The coloring of maps can also be stated in terms of graph theory, by considering it in terms of constructing a graph coloring of the planar
Four_color_theorem
Graph coloring variant in graph theory
In graph theory, a packing coloring (also called a broadcast coloring) is a type of graph coloring where vertices are assigned colors (represented by
Packing_coloring
Area of discrete mathematics
be formalized as asking for the crossing number of a complete bipartite graph. A graph coloring is a methodical assignment of labelling the elements of
Graph_theory
Unproven generalization of the four-color theorem
G {\displaystyle G} to the complete graph K k {\displaystyle K_{k}} , then G {\displaystyle G} must have a vertex coloring with k − 1 {\displaystyle k-1}
Hadwiger conjecture (graph theory)
Hadwiger_conjecture_(graph_theory)
otherwise. This decision problem is NP-complete. The problem may be generalized to triangle-free edge coloring, finding an assignment of colors to the
Monochromatic_triangle
Partition of a graph's nodes into cliques
and coloring is a reduction that can be used to prove the NP-completeness of the clique cover problem from the known NP-completeness of graph coloring. Perfect
Clique_cover
Special type of graph coloring
In graph theory, oriented graph coloring is a special type of graph coloring. Namely, it is an assignment of colors to vertices of an oriented graph that
Oriented_coloring
Computer compiler optimization technique
graph coloring portion of the register allocation problem can be solved in linear time. What causes the general graph coloring problem to be NP-complete and
Register_allocation
Unsolved problem on graph coloring
problem is an unsolved problem on graph coloring in mathematics. It is an extension of the planar map coloring problem (solved by the four color theorem)
Earth–Moon_problem
Concept in graph theory
has an incidence coloring with a given number of colors is NP-complete. Specifically, Li and Tu showed in 2008 that it is NP-complete to determine whether
Incidence_(graph)
Complexity class
given matrix whose entries are 0 or 1? (See #P-completeness of 01-permanent.) How many graph colorings using k colors are there for a particular graph
♯P-complete
Graph divided into two independent sets
endpoints of differing colors, as is required in the graph coloring problem. In contrast, such a coloring is impossible in the case of a non-bipartite graph,
Bipartite_graph
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
Edge-colored graph matching where all edges have distinct colors
transversals of Latin squares. Denote by Kn,n the complete bipartite graph on n + n vertices. Every proper n-edge coloring of Kn,n corresponds to a Latin square of
Rainbow_matching
Song by Kander and Ebb
"My Coloring Book" is a song written by Fred Ebb and John Kander. First performed by Sandy Stewart in 1962 on the television program The Perry Como Kraft
My_Coloring_Book
Graph operation
given graph, starting from the complete graph Kk. A similar construction may be used for list coloring in place of coloring. For k = 3, every k-critical
Hajós_construction
Conjecture about coloring graphs
problem about graph coloring, named after Paul Erdős, Vance Faber, and László Lovász, who formulated it in 1972. It says: If k complete graphs, each having
Erdős–Faber–Lovász_conjecture
Subcoloring is as difficult to solve exactly as coloring, in the sense that (like coloring) it is NP-complete. More specifically, the problem of determining
Subcoloring
Structure-preserving correspondence between node-link graphs
(that is, has the complete graph K3 as a subgraph) is homomorphically equivalent to K3. This is because, on one hand, a 3-coloring of G is the same as
Graph_homomorphism
US company
privately held corporation providing caramel color, burnt sugar and natural colorings for the food and beverage industry, before being acquired in 2021 by Givaudan
D.D._Williamson
Graph able to be partitioned into multiple independent sets
theory, a k-partite graph may be given as input to a computation with its coloring already determined; this can happen when the sets of vertices in the graph
Multipartite_graph
a coloring has a repetitive path is in NP, so testing whether a coloring is nonrepetitive is in co-NP, and Manin showed that it is co-NP-complete. The
Thue_number
On coloring the edges of graphs
either class one or class two, is NP-complete, there is no hope for a polynomial-time algorithm for best edge coloring. However, already Vizing's original
Vizing's_theorem
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
bipartite graphs and complete multipartite graphs. The simplest example of a graph that is not well-colored is a four-vertex path. Coloring the vertices in
Well-colored_graph
Theorem in combinatorics
{\displaystyle n\times n} Latin square corresponds to a proper edge coloring of the complete bipartite graph K n , n {\displaystyle K_{n,n}} with n {\displaystyle
Dinitz_theorem
In graph theory, a sum coloring of a graph is a labeling of its vertices by positive integers, with no two adjacent vertices having equal labels, that
Sum_coloring
Statement in mathematical combinatorics
^{2}\to \{1,2,\dots ,k\}} be a k-coloring of the complete graph on N {\displaystyle \mathbb {N} } . It is a stable coloring iff ∀ n ∈ N {\displaystyle \forall
Ramsey's_theorem
Planar maps require at most five colors
v_{1}} without affecting the coloring of the rest of G ′ {\displaystyle G'} . This frees color 1 for v {\displaystyle v} completing the task. If on the contrary
Five_color_theorem
Animation technique in which frames are hand-drawn
soundtrack than it is to synchronize a soundtrack to pre-existing animation. A completed cartoon soundtrack will feature music, sound effects, and dialogue performed
Traditional_animation
Wireless networking standard
"On IEEE 802.11: Wireless LAN Technology". arXiv:1307.2661 [cs.NI]. "The complete family of wireless LAN standards: 802.11 a, b, g, j, n" (PDF). The Physical
Wi-Fi_6
Duality of graph colorings and orientations
the Gallai–Hasse–Roy–Vitaver theorem is a form of duality between the colorings of the vertices of a given undirected graph and the orientations of its
Gallai–Hasse–Roy–Vitaver theorem
Gallai–Hasse–Roy–Vitaver_theorem
Function in algebraic graph theory
graph theory, a branch of mathematics. It counts the number of graph colorings as a function of the number of colors and was originally defined by George
Chromatic_polynomial
Numerical invariant of graphs
Tree-depth may also be defined using a form of graph coloring. A centered coloring of a graph is a coloring of its vertices with the property that every connected
Tree-depth
1996 studio album by Fugees
chance. In early 1995, he gave them a $135,000 advance and granted them complete artistic control for a follow-up album. The group used the money for recording
The_Score_(album)
American multinational food company
Kellogg and Crayola teamed up to create a fruit flavored cereal with a coloring book on the box. Crispix Crunch: Caramel Nut Crunch, Cran-Vanilla Crunch
Kellogg's
Every triangle-free planar graph is 3-colorable
3-list-colorable. However, Grötzsch's theorem itself does not extend from coloring to list coloring: there exist triangle-free planar graphs that are not 3-list-colorable
Grötzsch's_theorem
American fantasy media franchise based on novels by George R. R. Martin
was published in October 2013. The Official A Game of Thrones Coloring Book, a coloring book released in 2015 featuring 45 original black and white illustrations
A Song of Ice and Fire (franchise)
A_Song_of_Ice_and_Fire_(franchise)
Fundamental color in color mixing
vermilion, orpiment (King's yellow), and Bergblau (azurite) in partially complete colorings of planes in his solid. Johann Heinrich Lambert (a Swiss mathematician
Primary_color
Marvel Comics superhero
film Spider-Man has appeared in comics, cartoons, films, video games, coloring books, novels, records, children's books, and theme park rides. On television
Spider-Man
Undirected graph
that its deletion would decrease the number of colors needed in a graph coloring of the given graph. Each time a single edge or vertex (along with its incident
Critical_graph
vertex. These 11 uniform tilings have 32 different uniform colorings. A uniform coloring allows identical sided polygons at a vertex to be colored differently
List of Euclidean uniform tilings
List_of_Euclidean_uniform_tilings
Measurement of graph sparsity
least n {\displaystyle n} such that any two-edge-coloring of an n {\displaystyle n} -vertex complete graph must contain a monochromatic copy of G {\displaystyle
Degeneracy_(graph_theory)
Asynchrony and awaiting feature in programming languages
asynchronous libraries and APIs, an issue often referred to as "function coloring". Alternatives to async/await that do not suffer from this issue are called
Async/await
Type of decision problem in computer science
time, maintaining at each step a valid 4-coloring, is PSPACE-complete, even though the same problem for 3-colorings can be solved in polynomial time. Another
PSPACE-complete
Logic-based number-placement puzzle
be expressed as a graph coloring problem. The aim is to construct a 9-coloring of a particular graph, given a partial 9-coloring. The fewest clues possible
Sudoku
Non-crossing graph with vertices on outer face
Chvátal's art gallery theorem by Fisk (1978). A 3-coloring may be found in linear time by a greedy coloring algorithm that removes any vertex of degree at
Outerplanar_graph
Comic book reprints by Fantagraphics
The pages are recolored by Rich Tommaso, using the original comics as a coloring guide, unlike some of Fantagraphics' more scholarly reprints, as the books
The Complete Carl Barks Disney Library
The_Complete_Carl_Barks_Disney_Library
Family of graphs with 2n nodes and n(n-1) edges
pronic number n(n − 1). Its achromatic number is n: one can find a complete coloring by choosing each pair {ui, vi} as one of the color classes. Crown
Crown_graph
American fantasy drama TV series (2011–2019)
actors, such as Jack Gleeson and Sophie Turner, received frequent hair coloring. For characters such as Daenerys (Clarke) and her Dothraki, their hair
Game_of_Thrones
Victor Lage; Soares, Ronan; Sampaio, Rudini (2020). "PSPACE-completeness of two graph coloring games". Theoretical Computer Science. 824–825: 36–45. doi:10
List of PSPACE-complete problems
List_of_PSPACE-complete_problems
Chemical element with atomic number 27 (Co)
cobalus) has been cited by chemist Peter Wothers on this topic. "New and complete dictionary of the German language for Englishmen" s.v. "Das Wetter": "4
Cobalt
Influence of local substructure of a graph on global properties
graph has a coloring with a prescribed number of colors is known to be NP-hard. In addition to vertex coloring, other types of coloring are also studied
Extremal_graph_theory
The blow book, better known as a magic coloring book in modern variations, is a classic magic trick that has been performed for hundreds of years. It was
Blow_book
Mathematical conjecture
whose shortest synchronizing word has length exactly (n − 1)2. The road coloring problem is the problem of labeling the edges of a regular directed graph
Synchronizing_word
American photographer and environmentalist (1902–1984)
mood of a magical summer afternoon". For a short time Adams used hand-coloring, but declared in 1923 that he would do this no longer. By 1925 he had rejected
Ansel_Adams
Conjecture in graph theory
notion of graph coloring, since it follows from definitions that a k-coloring is the same as a Kk-coloring (a homomorphism into the complete graph on k vertices)
Hedetniemi's_conjecture
DC Comics imprint
Green Lantern is writer Al Ewing's first series for DC and will be a "complete reimagining" of the concept, starring multiple Lanterns". GamesRadar+.
Absolute_Universe
Pencil and paper map-coloring game
game, specifically a map-coloring game, involving the shading of areas in a line drawing according to the rules of graph coloring. With each move, the graph
Col_(game)
Cubic graph with 10 vertices and 15 edges
(possibly improper) coloring that breaks all of its symmetries; that is, its distinguishing number is three. Except for the complete graphs, it is the only
Petersen_graph
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