Graph coloring
In
graph theory, graph coloring is an assignment of
"colors", , to certain objects in a graph. Such objects can be vertices, edges, faces, or a mixture of those.
Vertex coloring is the most important kind. It is the starting point of the entire subject, and other coloring problems can be transformed into a vertex version. For example, an
edge coloring of a graph is just the vertex coloring of its
line graph. Likewise, a
face coloring of a
planar graph is just the vertex coloring of its
dual. However, to keep things in their perspective, non-vertex coloring problems are usually stated and studied
as are.
Encyclopedia
In
graph theory,
graph coloring is an assignment of
"colors", , to certain objects in a graph. Such objects can be vertices, edges, faces, or a mixture of those.
Vertex coloring is the most important kind. It is the starting point of the entire subject, and other coloring problems can be transformed into a vertex version. For example, an
edge coloring of a graph is just the vertex coloring of its
line graph. Likewise, a
face coloring of a
planar graph is just the vertex coloring of its
dual. However, to keep things in their perspective, non-vertex coloring problems are usually stated and studied
as are.
Graph coloring is not to be confused with graph labeling, which is an assignment of
labels, usually also in the form of numbers, to vertices or edges.
In graph coloring problems, colors are nothing more than markers for keeping track of adjacency or incidence.
In graph labeling problems, labels are calculable values that need to satisfy a certain numerical condition in the definition of the labeling.
Graph coloring enjoys many practical applications as well as theoretical challenges. Beside the classical types of problems, different limitations can also be set on the graph, or on the way a color is assigned, or even on the color itself. It has even reached popularity with the general public in the form of the popular number puzzle
Sudoku. Graph coloring is still a very active field of research.
Note: Many terms used in this article are defined in the glossary of graph theory.Vertex coloring
When used without any qualification, a
coloring of a
graph is always assumed to be a
vertex coloring, namely an assignment of colors to the vertices of the graph.
Again, when used without any qualification, a coloring is nearly always assumed to be
proper, meaning no two
adjacent vertices are assigned the same color.
Here, "adjacent" means sharing the same edge.
A coloring using at most
k colors is called a
k-coloring and is equivalent to the problem of partitioning the vertex set into
k or fewer independent sets.
The problem of coloring a graph has found a number of applications such as scheduling, register allocation in
compilers, frequency assignment in mobile radios, and pattern matching.
Chromatic number
The least number of colors needed to color the graph is called its
chromatic number χ.
For example the chromatic number of a complete graph of
n vertices , is .
A graph that can be assigned a
k-coloring is
k-colorable, and it is
k-chromatic if its chromatic number is exactly
k.
The problem of finding a minimum coloring of a graph is NP-hard.
The corresponding decision problem is
NP-complete, and in fact was one of Karp's 21 NP-complete problems. It remains NP-complete even on planar graphs of degree at most 4, as shown by Garey and Johnson in 1974, although on planar graphs it's trivial for
k ≠ 3 . There are however some efficient approximation algorithms that employ semidefinite programming.
Some properties of χ:
- χ = 1 if and only if G is totally disconnected.
- χ ≥ 3 if and only if G has an odd cycle .
- χ ≥ ω.
- χ ≤ Δ+1.
- χ ≤ Δ unless G is a complete graph or an odd cycle .
- χ ≤ 4, for any planar graph
...
G. This famous result, called the
four-color theorem, was stated by P. J. Heawood in 1890 , but remained unproven until 1976, when it was established by Kenneth Appel and Wolfgang Haken at the
University of Illinois at Urbana-Champaign.
Here Δ is the
maximum degree, and ω, the
clique number.
About infinite graphs, much less is known.
The following is one of the few results about infinite graph coloring:
- If all finite subgraphs of an infinite graph G are k-colorable, then so is G.
Chromatic polynomial
The
chromatic polynomial counts the number of ways a graph can be coloured using no more than a given number of colours. For example, using three colours, the graph in the image to the right can be coloured in 12 ways. With only two colours, it cannot be coloured at all. With four colours, it can be coloured in 24 + 4·12 = 72 ways: using all four colours, there are 4! = 24 valid colourings ; and for every choice of three of the four colours, there are 12 valid 3-colourings. So, for the graph in the example, a table of the number of valid colourings would start like this:
| Available colors | 1 | 2 | 3 | 4 | … |
| Number of colorings | 0 | 0 | 12 | 72 | … |
The chromatic polynomial is a function that counts the number of -colorings of . As the name indicates, for a given the function is indeed a
polynomial in . For the example graph, , and indeed .
The chromatic polynomial includes at least as much information about the colorability of as does the chromatic number. Indeed, χ is the smallest positive integer that is not a root of the chromatic polynomial,
-
It was first used by
Birkhoff and Lewis in their attack on the
four-color theorem.
It remains an unsolved problem to characterize graphs which have the same chromatic polynomial and to determine precisely what polynomials are chromatic.
Examples
Properties
-
- if contains an edge
- , if .
- is the number of acyclic orientations of
- If has vertices, edges, and components …,, then
- has degree .
- The coefficient of in is 1.
- The coefficient of in is .
- The coefficients of … are all zero.
- The coefficient of is nonzero.
- ?
- The coefficients of every chromatic polynomial alternate in signs.
- A graph G with vertices is a tree if and only .
- The derivative evaluated at unity, is the chromatic invariant up to sign
Computing the chromatic polynomial
Whenever contains a loop, it cannot be legally colored at all, so
If not a loop, then the chromatic polynomial satisifes the recurrence relation
where is the graph with the edge removed, and is the graph with the edge's endpoints
contracted into a single vertex.
The two expressions give raise to a recursive procedure, called the deletion–contraction algorithm. In the recursive step, the instance is transformed into two instances that are at least one edge smaller, so the running time comes within a polynomial factor of , where is the number of edges. The analysis can be improved to .
Other algorithms are known, but all are exponential in the size of the graph. For some classes of graphs, polynomial-time algorithms are known. These include the empty and complete graphs, forests, chordal graphs, wheels, and ladders. In general, computing the chromatic polynomial is Sharp-P-complete, so it is unlikely that a polynomial time algorithm for all graphs will be found.
List of other colorings
Not only can the idea of vertex coloring be extended to edges, but also be added with different conditions to form new structures and problems.
- edge coloring - edges are colored
- list coloring - each vertex chooses from a list of colors
- list edge-coloring - each edge chooses from a list of colors
- total coloring - vertices and edges are colored
- harmonious coloring - every pair of colors appears on at most one edge
- complete coloring - every pair of colors appears on at least one edge
- exact coloring - every pair of colors appears on exactly one edge
- acyclic coloring - every 2-chromatic subgraph is acyclic
- strong coloring - every color appears in every partition of equal size exactly once
- on-line coloring - the instance of the problem is not given in advance and its successive parts become known over time
- equitable coloring - the sizes of color classes differ by at most one
- sum-coloring - the criterion of minimalization is the sum of colors
- T-coloring - distance between two colors of adjacent vertices must not belong to fixed set T
- rank coloring - if two vertices have the same color i, then every path between them contain a vertex with color greater than i.
- interval edge-coloring - a color of edges meeting in a common vertex must be contiguous.
- circular coloring - motivated by task systems in which production proceeds in a cyclic way
- path coloring - models a routing problem in graphs
- fractional coloring - vertices may have multiple colors, and on each edge the sum of the color parts of each vertex is not greater than one
Some improper colorings:
- cocoloring -- every color class induces an independent set or a clique
- subcoloring -- every color class induces a union of cliques
References
Literature
- R. L. Brooks: On colouring the nodes of a network. In: Proc. Cambridge Phil. Soc. 37, 1941, 194–197.
- N. G. de Bruijn, P. Erdős: A colour problem for infinite graphs and a problem in the theory of relations. In: Nederl. Akad. Wetensch. Proc. Ser. A 54, 1951 371–373.
- Tommy R. Jensen, Bjarne Toft: Graph coloring problems. Wiley-Interscience, New York 1995 ISBN 0-471-02865-7.
- Marek Kubale and others: Graph Colorings. American Mathematical Society 2004 ISBN 0-8218-3458-4.
- M. R. Garey, D. S. Johnson, and L. Stockmeyer. . Proceedings of the sixth annual ACM symposium on Theory of computing, p.47-63. 1974.
A1.1: GT4, pg.191.
See also
External links
- by Joseph Culberson
- by Jim Andrews and Mike Fellows is a graph coloring puzzle
- GF-Graph Coloring Program
