Graph coloring

A proper vertex coloring of the Petersen graph with 3 colors, the minimum number possible.

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 constraints, such as that no two adjacent elements have the same color. Graph coloring is a special case of graph labeling. In its simplest form, it is a way of coloring the vertices of a graph such that no two adjacent vertices are of the same color; this is called a 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 planar graph assigns a color to each face (or region) so that no two faces that share a boundary have the same color.

Vertex coloring is often used to introduce graph coloring problems, since other coloring problems can be transformed into a vertex coloring instance. For example, an edge coloring of a graph is just a vertex coloring of its line graph, and a face coloring of a plane graph is just a vertex coloring of its dual. However, non-vertex coloring problems are often stated and studied as-is. This is partly pedagogical, and partly because some problems are best studied in their non-vertex form, as in the case of edge coloring.

The convention of using colors originates from coloring the countries in a political map, where each face is literally colored. This was generalized to coloring the faces of a graph embedded in the plane. By planar duality it became coloring the vertices, and in this form it generalizes to all graphs. In mathematical and computer representations, it is typical to use the first few positive or non-negative integers as the "colors". In general, one can use any finite set as the "color set". The nature of the coloring problem depends on the number of colors but not on what they are.

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 Glossary of graph theory.

A map of the United States using colors to follow the four color theorem.

The first results about graph coloring deal almost exclusively with planar graphs in the form of map coloring. While trying to color a map of the counties of England in 1852, Francis Guthrie postulated the four color conjecture, noting that four colors were sufficient to color the map so that no regions sharing a common border received the same color. Guthrie's brother Frederick passed on the question to his mathematics teacher Augustus De Morgan at University College, who mentioned it in a letter to William Hamilton in 1852. Arthur Cayley raised the problem at a meeting of the London Mathematical Society in 1879. The same year, Alfred Kempe published a paper that claimed to establish the result, and for a decade the four color problem was considered solved. For his accomplishment Kempe was elected a Fellow of the Royal Society and later President of the London Mathematical Society.

In 1890, Percy John Heawood pointed out that Kempe's argument was wrong. However, in that paper he proved the five color theorem, saying that every planar map can be colored with no more than five colors, using ideas of Kempe. In the following century, a vast amount of work was done and theories were developed to reduce the number of colors to four, until the four color theorem was finally proved in 1976 by Kenneth Appel and Wolfgang Haken. The proof went back to the ideas of Heawood and Kempe and largely disregarded the intervening developments. The proof of the four color theorem is noteworthy, aside from its solution of a century-old problem, for being the first major computer-aided proof.

In 1912, George David Birkhoff introduced the chromatic polynomial to study the coloring problem, which was generalised to the Tutte polynomial by W. T. Tutte, both of which are important invariants in algebraic graph theory. Kempe had already drawn attention to the general, non-planar case in 1879, and many results on generalisations of planar graph coloring to surfaces of higher order followed in the early 20th century.

In 1960, Claude Berge formulated another conjecture about graph coloring, the strong perfect graph conjecture, originally motivated by an information-theoretic concept called the zero-error capacity of a graph introduced by Shannon. The conjecture remained unresolved for 40 years, until it was established as the celebrated strong perfect graph theorem by Chudnovsky, Robertson, Seymour, and Thomas in 2002.

Graph coloring has been studied as an algorithmic problem since the early 1970s: the chromatic number problem (see section § Vertex coloring below) is one of Karp's 21 NP-complete problems from 1972, and at approximately the same time various exponential-time algorithms were developed based on backtracking and on the deletion-contraction recurrence of Zykov (1949). One of the major applications of graph coloring, register allocation in compilers, was introduced in 1981.

This graph can be 3-colored in 12 different ways.

When used without any qualification, a coloring of a graph almost always refers to a proper vertex coloring, namely a labeling of the graph's vertices with colors such that no two vertices sharing the same edge have the same color. Since a vertex with a loop (i.e. a connection directly back to itself) could never be properly colored, it is understood that graphs in this context are loopless.

The terminology of using colors for vertex labels goes back to map coloring. Labels like red and blue are only used when the number of colors is small, and normally it is understood that the labels are drawn from the integers {1, 2, 3, ...}.

A coloring using at most G is called its chromatic number, and is often denoted χ(G) is also used to denote the Euler characteristic of a graph. A graph that can be assigned a (proper) k-chromatic if its chromatic number is exactly k independent sets, and the terms E3 (red) admits a 1-coloring; the complete graph

The chromatic polynomial counts the number of ways a graph can be colored using some of a given number of colors. For example, using three colors, the graph in the adjacent image can be colored in 12 ways. With only two colors, it cannot be colored at all. With four colors, it can be colored in 24 + 4 × 12 = 72 ways: using all four colors, there are 4! = 24 valid colorings (every assignment of four colors to any 4-vertex graph is a proper coloring); and for every choice of three of the four colors, there are 12 valid 3-colorings. So, for the graph in the example, a table of the number of valid colorings would start like this:

Available colors	1	2	3	4	...
Number of colorings	0	0	12	72	...

The chromatic polynomial is a function G. As the name indicates, for a given P(G, t) = t(t − 1)(t − 2), and indeed χ is the smallest positive integer that is not a zero of the chromatic polynomial 0}}"}},"i":0}}]}'/>χ(G) = min{k : P(G, k) > 0}.

Chromatic polynomials for certain graphs

Triangle Kn

t(t − 1)

Cycle An edge coloring of a graph is a proper coloring of the edges, meaning an assignment of colors to edges so that no vertex is incident to two edges of the same color. An edge coloring with k matchings. The smallest number of colors needed for an edge coloring of a graph G is the fewest colors needed in any total coloring of k-face-colorable then d vertices as a vector in 1≤χ(G)≤n. The only graphs that can be 1-colored are edgeless graphs. A complete graph χ-bounded if there is some function ℱ can be colored with at most G. For the family of the perfect graphs this function is χ(G)=n and Δ(G)=2, so for these graphs this bound is best possible. In all other cases, the bound can be slightly improved; Brooks' theorem states that Brooks' theorem: (i,j) is not an edge in λmax(W),λmin(W) are the largest and smallest eigenvalues of W as above. Then: Wi,j≤−1k−1 whenever χV(G) to be the least k for which such a matrix ϑ(G¯)≤χ(G). Fractional chromatic number: The fractional chromatic number of a graph is a lower bound on the chromatic number as well: 1947, AlexanderZykov 1949, JanMycielski 1955): There exist triangle-free graphs with arbitrarily high chromatic number. To prove this, both, Mycielski and Zykov, each gave a construction of an inductively defined family of triangle-free graphs but with arbitrarily large chromatic number. Burling (1965) constructed axis aligned boxes in ℝ2 are not χ-bounded. From Brooks's theorem, graphs with high chromatic number must have high maximum degree. But colorability is not an entirely local phenomenon: A graph with high girth looks locally like a tree, because all cycles are long, but its chromatic number need not be 2: Theorem (Erdős): There exist graphs of arbitrarily high girth and chromatic number. Bounds on the chromatic index edit An edge coloring of G is a vertex coloring of its line graph Δ or (Fawcett 1978). Open problems edit As stated above, G, the polynomial [5,∞) and that kn assignments of k colors to n vertices and checks for each if it is legal. To compute the chromatic number and the chromatic polynomial, this procedure is used for every O(1.7272n), respectively. Exponentially faster algorithms are also known for 5- and 6-colorability, as well as for restricted families of graphs, including sparse graphs. Contraction edit The contraction G+uv is the graph with the edge G−uv is the graph with the edge v1, ..., vi's neighbours among χ(G) colors. On the other hand, greedy colorings can be arbitrarily bad; for example, the crown graph on n vertices can be 2-colored, but has an ordering that leads to a greedy coloring with O((n+m)log⁡n) where View original on Wikipedia | Back | FazBrowse Home