In
graph theory, a complete coloring is a
vertex coloring 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, which requires every pair of colors to appear on ''at most'' one pair of adjacent vertices.
Complexity theory
Finding is an
optimization problem. The
decision problem for complete coloring can be phrased as:
:INSTANCE: a graph and positive integer
:QUESTION: does there exist a
partition 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, 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
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
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 (that is, graphs having no independent set of more than two vertices),
s and
interval graphs, 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 is known to be proportional to
, but the constant of proportionality is not known precisely.
[.]
References
External links
A compendium of NP optimization problemsby Keith Edwards
{{DEFAULTSORT:Complete Coloring
Graph coloring