graph theory In mathematics and computer science, graph theory is the study of ''graph (discrete mathematics), graphs'', which are mathematical structures used to model pairwise relations between objects. A graph in this context is made up of ''Vertex (graph ...

, a cocoloring of a graph ''G'' is an assignment of

color Color (or colour in English in the Commonwealth of Nations, Commonwealth English; American and British English spelling differences#-our, -or, see spelling differences) is the visual perception based on the electromagnetic spectrum. Though co ...

s to the vertices such that each color class forms an independent set in ''G'' or in the

complement Complement may refer to: The arts * Complement (music), an interval that, when added to another, spans an octave ** Aggregate complementation, the separation of pitch-class collections into complementary sets * Complementary color, in the visu ...

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 mathematics, mathematical field of graph theory, a bipartite graph (or bigraph) is a Graph (discrete mathematics), graph whose vertex (graph theory), vertices can be divided into two disjoint sets, disjoint and Independent set (graph theo ...

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 , where they called these ...

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 methodic assignment of labels traditionally called "colors" to elements of a graph. The assignment is subject to certain constraints, such as that no two adjacent elements have the same color. Graph coloring i ...

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

*. *. *. *. *. *. {{refend Graph coloring