graph theory In mathematics, graph theory is the study of ''graphs'', which are mathematical structures used to model pairwise relations between objects. A graph in this context is made up of '' vertices'' (also called ''nodes'' or ''points'') which are conne ...

, a complete coloring is a

vertex coloring 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 ...

in which every pair of colors appears on ''at least'' one pair of adjacent vertices. Equivalently, a complete coloring is minimal in the sense that it cannot be transformed into a proper coloring with fewer colors by merging pairs of color classes. The achromatic number of a graph is the maximum number of colors possible in any complete coloring of . A complete coloring is the opposite of a

harmonious coloring In graph theory, a harmonious coloring is a (proper) vertex coloring in which every pair of colors appears on ''at most'' one pair of adjacent vertices. It is the opposite of the complete coloring, which instead requires every color pairing to o ...

, which requires every pair of colors to appear on ''at most'' one pair of adjacent vertices.

Complexity theory

Finding is an

optimization problem In mathematics, computer science and economics, an optimization problem is the problem of finding the ''best'' solution from all feasible solutions. Optimization problems can be divided into two categories, depending on whether the variables ...

. The

decision problem In computability theory and computational complexity theory, a decision problem is a computational problem that can be posed as a yes–no question of the input values. An example of a decision problem is deciding by means of an algorithm wheth ...

for complete coloring can be phrased as: :INSTANCE: a graph and positive integer :QUESTION: does there exist a

partition Partition may refer to: Computing Hardware * Disk partitioning, the division of a hard disk drive * Memory partition, a subdivision of a computer's memory, usually for use by a single job Software * Partition (database), the division of a ...

of into or more

disjoint sets In mathematics, two sets are said to be disjoint sets if they have no element in common. Equivalently, two disjoint sets are sets whose intersection is the empty set.. For example, and are ''disjoint sets,'' while and are not disjoint. A ...

such that each is an independent set for and such that for each pair of distinct sets is not an independent set. Determining the achromatic number is

NP-hard In computational complexity theory, NP-hardness ( non-deterministic polynomial-time hardness) is the defining property of a class of problems that are informally "at least as hard as the hardest problems in NP". A simple example of an NP-hard pr ...

; determining if it is greater than a given number is

NP-complete In computational complexity theory, a problem is NP-complete when: # it is a problem for which the correctness of each solution can be verified quickly (namely, in polynomial time) and a brute-force search algorithm can find a solution by tryi ...

, as shown by Yannakakis and Gavril in 1978 by transformation from the minimum maximal matching problem. A1.1: GT5, pg.191. Note that any coloring of a graph with the minimum number of colors must be a complete coloring, so minimizing the number of colors in a complete coloring is just a restatement of the standard

graph coloring 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 o ...

problem.

Algorithms

For any fixed ''k'', it is possible to determine whether the achromatic number of a given graph is at least ''k'', in linear time. The optimization problem permits approximation and is approximable within a

O\left(, V, /\sqrt\right)

approximation ratio An approximation is anything that is intentionally similar but not exactly equal to something else. Etymology and usage The word ''approximation'' is derived from Latin ''approximatus'', from ''proximus'' meaning ''very near'' and the prefix ' ...

Special classes of graphs

The NP-completeness of the achromatic number problem holds also for some special classes of graphs:

bipartite graph In the mathematical field of graph theory, a bipartite graph (or bigraph) is a graph whose vertices can be divided into two disjoint and independent sets U and V, that is every edge connects a vertex in U to one in V. Vertex sets U and V are ...

s,. complements of

s (that is, graphs having no independent set of more than two vertices),

cograph In graph theory, a cograph, or complement-reducible graph, or ''P''4-free graph, is a graph that can be generated from the single-vertex graph ''K''1 by complementation and disjoint union. That is, the family of cographs is the smallest class of ...

s and

interval graph In graph theory, an interval graph is an undirected graph formed from a set of intervals on the real line, with a vertex for each interval and an edge between vertices whose intervals intersect. It is the intersection graph of the intervals. In ...

s, and even for trees. For complements of trees, the achromatic number can be computed in polynomial time. For trees, it can be approximated to within a constant factor. The achromatic number of an ''n''-dimensional

hypercube graph In graph theory, the hypercube graph is the graph formed from the vertices and edges of an -dimensional hypercube. For instance, the cube graph is the graph formed by the 8 vertices and 12 edges of a three-dimensional cube. has vertices, ...

is known to be proportional to

\sqrt

, but the constant of proportionality is not known precisely..

References

External links

A compendium of NP optimization problems

by Keith Edwards {{DEFAULTSORT:Complete Coloring Graph coloring