In
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 conn ...
, a cocoloring of a graph ''G'' is an assignment of
color
Color (American English) or colour (British English) is the visual perceptual property deriving from the spectrum of light interacting with the photoreceptor cells of the eyes. Color categories and physical specifications of color are assoc ...
s to the vertices such that each color class forms an
independent set in ''G'' or in the
complement
A complement is something that completes something else.
Complement may refer specifically to:
The arts
* Complement (music), an interval that, when added to another, spans an octave
** Aggregate complementation, the separation of pitch-clas ...
of ''G''. The cochromatic number z(''G'') of ''G'' is the fewest colors needed in any cocolorings of ''G''. The graphs with cochromatic number 2 are exactly the
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 ar ...
s, complements of bipartite graphs, and
split graph
In graph theory, a branch of mathematics, a split graph is a graph in which the vertices can be partitioned into a clique and an independent set. Split graphs were first studied by , and independently introduced by .
A split graph may have m ...
s.
As the requirement that each color class be a clique or independent is weaker than the requirement for
coloring (in which each color class must be an independent set) and stronger than for
subcoloring (in which each color class must be a disjoint union of cliques), it follows that the cochromatic number of ''G'' is less than or equal to the
chromatic number
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 ...
of ''G'', and that it is greater than or equal to the subchromatic number of ''G''.
Cocoloring was named and first studied by . characterizes critical 3-cochromatic graphs, while describe algorithms for approximating the cochromatic number of a graph. defines a class of ''perfect cochromatic graphs'', analogous to the definition of perfect graphs via graph coloring, and provides a forbidden subgraph characterization of these graphs.
References
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Graph coloring