Complete coloring

File:Complete coloring clebsch graph.svg

Complete coloring of the Clebsch graph with 8 colors. Every pair of colors appears on at least one edge. No complete coloring with more colors exists: in any 9-coloring some color would appear only at one vertex, and there would not be enough neighboring vertices to cover all pairs involving that color. Therefore, the achromatic number of the Clebsch graph is 8.

In graph theory, a complete coloring is a (proper) 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 Template:Math of a graph Template:Mvar is the maximum number of colors possible in any complete coloring of Template:Mvar.

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 Template:Math is an optimization problem. The decision problem for complete coloring can be phrased as:

INSTANCE: a graph Template:Math and positive integer Template:Mvar

QUESTION: does there exist a partition of Template:Mvar into Template:Mvar or more disjoint sets Template:Math such that each Template:Mvar is an independent set for Template:Mvar and such that for each pair of distinct sets Template:Math is not an independent set.

Determining the achromatic number is NP-hard; 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.<ref name="gj">Template:Citation A1.1: GT5, pg.191.</ref>

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 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.<ref name="fhhm"/>

The optimization problem permits approximation and is approximable within a <math>O\left(|V|/\sqrt{\log |V|}\right)</math> approximation ratio.<ref name="cv">Template:Citation.</ref>

Special classes of graphs

The NP-completeness of the achromatic number problem holds also for some special classes of graphs: bipartite graphs,<ref name="fhhm">Template:Citation.</ref> complements of bipartite graphs (that is, graphs having no independent set of more than two vertices),<ref name="gj"/> cographs and interval graphs,<ref>Template:Citation.</ref> and even for trees.<ref>Template:Citation.</ref>

For complements of trees, the achromatic number can be computed in polynomial time.<ref>Template:Citation.</ref> For trees, it can be approximated to within a constant factor.<ref name="cv"/>

The achromatic number of an n-dimensional hypercube graph is known to be proportional to <math>\sqrt{n2^n}</math>, but the constant of proportionality is not known precisely.<ref>Template:Citation.</ref>

External links

• A compendium of NP optimization problems
• A Bibliography of Harmonious Colourings and Achromatic Number by Keith Edwards