Vertex_chromatic

Vertex chromatic number

A 3-coloring suits this graph, but fewer colors would result in adjacent vertices of the same color. Finding the minimum number of colors for an arbitrary graph is NP-hard.

In graph theory, graph coloring is a special case of graph labeling; it is an assignment of labels traditionally called "colors" to elements of a graph subject to certain constraints. In its simplest form, it is a way of coloring the vertices of a graph such that no two adjacent vertices share 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 share 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 the starting point of the subject, and other coloring problems can be transformed into a vertex version. For example, an edge coloring of a graph is just a vertex coloring of its line graph, and a face coloring of a planar graph is just a vertex coloring of its planar dual. However, non-vertex coloring problems are often stated and studied as is. That is partly for perspective, and partly because some problems are best studied in non-vertex form, as for instance is edge coloring.

The convention of using colors originates from coloring the countries of a 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 nonnegative 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 the glossary of graph theory.

History

See also: History of the four color theorem and History of graph theory

The first results about graph coloring deal almost exclusively with planar graphs in the form of the coloring of maps. While trying to color a map of the counties of England, 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 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. Over a century of failed attempts to establish the conjecture followed, until the four colour theorem was finally proved in 1976, a result that is also noteworthy for being the first major computer-aided proof.

In 1912, Birkhoff introduced the chromatic polynomial to study the coloring problems, which was generalised to the Tutte polynomial by Tutte, important structures in algebraic graph theory. Kempe had already drawn attention to the general, non-planar case in 1879¹, and many results on generalisations of planar graph coloring to surfaces of higher order followed in the early 20th century.

Graph coloring has been studied as an algorithmic problem since the early 1970s. The chromatic number problem is one of Karp’s 21 NP-complete problems from 1972, around the time of various exponential-time algorithms based on backtracking and Zykov’s deletion--contraction recurrence from 1949². One of the major applications of graph coloring, register allocation in compilers was introduced in 1981.

Variants

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 (proper) k-coloring and is equivalent to the problem of partitioning the vertex set into k or fewer independent sets.

Chromatic number

The least number of colors needed to color a graph G is called its chromatic number, χ(G). For example the chromatic number of a complete graph $K n$ of n vertices (a graph with an edge between every two vertices), is $χ(K n) = n$ . A graph that can be assigned a (proper) k-coloring is k-colorable, and it is k-chromatic if its chromatic number is exactly k.

Some properties of χ(G):

χ(G) = 1 if and only if G is totally disconnected, i.e., it has no edges.
χ(G) ≥ 3 if and only if G has an odd cycle (equivalently, if G is not bipartite).
χ(G) ≥ ω(G), the clique number, i.e., the number of vertices of a largest clique in G.
χ(G) ≤ Δ(G) + 1, 1 more than the maximum degree of a vertex.
χ(G) ≤ Δ(G) for a connected, simple graph G, unless G is a complete graph or an odd cycle (Brooks' theorem).
χ(G) ≤ 4, for any planar graph G. This famous result, called the four-color theorem, was stated by Augustus De Morgan in 1852, but remained unproven until 1976, when it was established by Kenneth Appel and Wolfgang Haken.

For planar graphs, vertex colorings are essentially dual to nowhere-zero flows.

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. (de Bruijn and Erdős 1951)

The chromatic number of the plane, where two points are adjacent if they have unit distance, is unknown, although it is one of 4, 5, 6, or 7. Other open problems concerning the chromatic number of graphs include the Hadwiger conjecture stating that every graph with chromatic number k has a complete graph on k vertices as a minor, and the Erdős–Faber–Lovász conjecture bounding the chromatic number of unions of complete graphs that have at exactly one vertex in common to each pair.

Chromatic polynomial

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

The chromatic polynomial counts the number of ways a graph can be colored using no more than a given number of colors. For example, using three colors, the graph in the image to the right 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 $P (G, t)$ that counts the number of $t$ -colorings of $G$ . As the name indicates, for a given $G$ the function is indeed a polynomial in $t$ . For the example graph, $P (G, t) = t (t - 1) 2 (t - 2)$ , and indeed $P (G,4) = 72$ .

The chromatic polynomial includes at least as much information about the colorability of $G$ as does the chromatic number. Indeed, χ is the smallest positive integer that is not a root of the chromatic polynomial,

χ(G) = min{k : P (G, k) > 0}

It was first used by Birkhoff and Lewis in their attack on the four-color theorem. They conjectured that, for a planar graph $G$ , $P (G, t)$ has no zeros in the region $[4,\infty)$ . Although it is known that such a chromatic polynomial has no zeros in the region $[5,\infty)$ and that $P(G,4) \neq 0$ . Their conjecture is still unresolved.

It remains an unsolved problem to characterize graphs which have the same chromatic polynomial and to determine precisely which polynomials are chromatic.

Examples

The Petersen graph has chromatic number 3.

Chromatic polynomials for certain graphs
Triangle $K 3$	$t (t - 1)(t - 2)$
Complete graph $K n$	$t(t-1)(t-2) \cdots (t-(n-1))$
Tree with $n$ vertices	$t (t - 1) n - 1$
Cycle $C n$	$(t - 1) n + ( - 1) n (t - 1)$
Petersen graph	$t (t - 1)(t - 2)(t 7 - 12 t 6 + 67 t 5 - 230 t 4 + 529 t 3 - 814 t 2 + 775 t - 352)$

Properties

$P (G,0) = 0$
$P (G,1) = 0$ if $G$ contains an edge
$P (G, t) = 0$ , if $t < χ(G)$ .
$( - 1) | V (G) | P (G, - 1)$ is the number of acyclic orientations of $G$ ³
If G has n vertices, m edges, and k components G₁,G₂,…,G_k, then
- $P (G, t)$ has degree $n$ .
- The coefficient of $t n$ in $P (G, t)$ is 1.
- The coefficient of $t n - 1$ in $P (G, t)$ is $- m$ .
- The coefficients of $t^0, t^1, \dots, t^{k-1}$ are all zero.
- The coefficient of $t k$ is nonzero.
- $P(G, t) = P(G_1, t)P(G_2,t) \cdots P(G_k,t)$
The coefficients of every chromatic polynomial alternate in signs.
A graph G with $n$ vertices is a tree if and only if $P (G, t) = t (t - 1) n - 1$ .
The derivative evaluated at unity, $P'(G,1)$ is the chromatic invariant $θ(G)$ up to sign

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
Strong edge coloring	Edges are colored such that each color class induces a matching (equivalent to coloring the square of the line graph)
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
Oriented coloring	Takes into account orientation of edges of the graph

Some improper colorings:

Cocoloring	Every color class induces an independent set or a clique
Subcoloring	Every color class induces a union of cliques
Weak coloring	Every non-isolated node has at least one neighbor with a different color

Generalizations

Coloring can be considered for signed graphs and gain graphs. Similar questions arise.

Algorithms

Graph coloring is computationally hard. It is NP-complete to decide if a given graph admits a coloring which uses at most a given number of colors. The problem remains NP-complete even on planar graphs of degree at most 4 ⁴.

Efficient algorithms

Determining if a graph can be colored with $2$ colors is equivalent to determining whether or not the graph is bipartite, and thus polynomial time-computable. As a consequence of the Strong Perfect Graph Theorem, the chromatic number of a perfect graph can be computed in polynomial time. Other classes of graphs that admit polynomial time algorithms are forests and chordal graphs.

Exact algorithms

Brute-force search for a $k$ -coloring considers every of the $\binom{n}{k}$ assignments of colors to vertices and checks for each if it is legal. To compute the chromatic number and the chromatic polynomial, this procedure is used for every $k$ , for a total running time of $O ((n + 1)!)$ , impractical for all but the smallest input graphs.

This can be improved based on Zykov’s identity: If $e$ is not a loop, then the chromatic number satisfies the recurrence relation:

$\chi(G) = \min \{ \, \chi(G+uv), \chi(G/uv)\,\}$

where $u$ and $v$ are nonadjecent vertices, $G + u v$ is the graph with the edge $u v$ added, and $G / u v$ is the graph where the two vertices are contracted into a single vertex.

The expressions gives rise to a recursive procedure, called the deletion–contraction algorithm. The running time satisfies the same recurrence relation as the Fibonacci numbers, so in the worst case, the algorithm runs in time within a polynomial factor of $((1+\sqrt{5})/2)^{n+m}=O(1.6180)^{n+m}$ ⁵. The analysis can be improved to within a polynomial factor of the number $t (G)$ of spanning trees of the input graph ⁶. In practice, branch and bound strategies and graph isomorphism rejection are employed to avoid some recursive calls, the running time depends on the heuristic used to pick the vertex pair.

Greedy colorings

Algorithm of Welsh and Powell

The Welsh-Powell algorithm⁷ for coloring a graph uses a simple heuristic improvement to a completely naive greedy algorithm. By processing the vertices in non-increasing order of degree, that is, ordering them so that $d_1 \geq d_2 \geq \cdots \geq d_n$ , where $d i$ is the degree of the $i$ th node, this algorithm uses $α(G)$ colors where

$\alpha(G) \leq \max_i \min \left\{ \begin{array}{c} d_i + 1 ,\\ i. \end{array} \right.$

The algorithm is as follows:

Sort the vertices in decreasing order of degree and initially have every vertex uncolored.
Traverse the vertices in order, giving a vertex color 1 if it is uncolored and it does not yet have a neighbor with color 1.
Repeat this process with colors 2, 3, etc. until no vertex is uncolored.

This algorithm does not necessarily find a $χ(G)$ coloring.

Nonexact algorithms

Here we can find the sequential algorithms (those that color one vertex at a time), heuristic algorithms, global randomized algorithms, metaheuristic algorithms (using simulated annealing, tabu search, etc.), and genetic algorithms, to name several types.

For k ≥ 3 the problem is NP-Complete. By the gap theorem, this implies that the problem can not be approximated by a polynomial algorithm within a factor of 4/3 unless P=NP.

Computing the chromatic polynomial

Other algorithms are known, but all are exponential in the size of the graph. In general, computing the chromatic polynomial is #P-complete, so it is unlikely that a polynomial time algorithm for all graphs will be found.

Applications

The problem of coloring a graph has found a number of applications. Some of them are scheduling, register allocation in compilers, frequency assignment in mobile radios, and pattern matching.

The recreational puzzle Sudoku can be seen as completing a 9-coloring a given specific graph on 81 vertices.

References

^ Jensen and Toft, p. 2
^ Zykov, 1949
^ Stanley, 1973
^ M. R. Garey, D. S. Johnson, and L. Stockmeyer. Some simplified NP-complete problems. Proceedings of the sixth annual ACM symposium on Theory of computing, p.47-63. 1974.
^ H. S. Wilf, Algorithms and Complexity, Prentice–Hall, 1986.
^ K. Sekine, H. Imai, S. Tani, Computing the Tutte polynomial of a graph of moderate size, Algorithms and Computation, 6th International Symposium (ISAAC ’95), Cairns, Australia, December 4–6, 1995, Lecture Notes in Computer Science 1004, Springer, 1995, pp. 224–233.
^ D. J. A. Welsh and M. B. Powell (1967). "An upper bound for the chromatic number of a graph and its application to timetabling problems". The Computer Journal 10 (1): 85–86. doi:10.1093/comjnl/10.1.85.

Literature

R. L. Brooks (1941). "On colouring the nodes of a network". Proc. Cambridge Phil. Soc. 37: 194–197.
N. G. de Bruijn; P. Erdős (1951), "A colour problem for infinite graphs and a problem in the theory of relations", Nederl. Akad. Wetensch. Proc. Ser. A 54: 371–373 (= Indag. Math. 13)
Tommy R. Jensen and Bjarne Toft (1995), Graph Coloring Problems, Wiley-Interscience, New York, ISBN 0-471-02865-7
Marek Kubale et al. (2004). Graph Colorings, American Mathematical Society. ISBN 0-8218-3458-4.
Michael R. Garey, David S. Johnson, and L. Stockmeyer (1974). "Some simplified NP-complete problems". Proceedings of the sixth annual ACM symposium on Theory of computing: 47-63.
Michael R. Garey and David S. Johnson (1979). Computers and Intractability: A Guide to the Theory of NP-Completeness, W.H. Freeman. ISBN 0-7167-1045-5. A1.1: GT4, pg.191.
West, Douglas B. (1996). Introduction to Graph Theory, Prentice-Hall. ISBN 0-13-227828-6.
A. A. Zykov (1949). "On some properties of linear complexes (in Russian)". Math. Sbornik. 24: 163-188.

External links

Graph Coloring Page by Joseph Culberson (graph coloring programs)
CoLoRaTiOn by Jim Andrews and Mike Fellows is a graph coloring puzzle
GF-Graph Coloring Program [1]
Links to Graph Coloring source codes
Code for efficiently computing Tutte, Chromatic and Flow Polynomials by Gary Haggard, David J. Pearce and Gordon Royle: [2]

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