Wikipedia
In graph theory, a subcoloring is an assignment of colors to a graph's vertices such that each color class induces a vertex disjoint union of cliques. That is, each color class should form a cluster graph.
A subchromatic number χ(G) of a graph G is the least number of colors needed in any subcoloring of G.
Subcoloring and subchromatic number were introduced by .
Every proper coloring and cocoloring of a graph are also subcolorings, so the subchromatic number of any graph is at most equal to the cochromatic number, which is at most equal to the chromatic number.
Subcoloring is as difficult to solve exactly as coloring, in the sense that (like coloring) it is NP-complete. More specifically, the problem of determining whether a graph has subchromatic number at most 2 is NP-complete, even for
- triangle-free planar graphs with maximum degree 4 ,
- comparability graphs ,
- planar perfect graphs with maximum degree 4 ,
- planar graphs with girth 5 .
The subchromatic number of a cograph can be computed in polynomial time . For every fixed integer r, it is possible to decide in polynomial time whether the subchromatic number of interval and permutation graphs is at most r .