[1] However, in some applications of graph theory, a k-partite graph may be given as input to a computation with its coloring already determined; this can happen when the sets of vertices in the graph represent different types of objects.
For instance, folksonomies have been modeled mathematically by tripartite graphs in which the three sets of vertices in the graph represent users of a system, resources that the users are tagging, and tags that the users have applied to the resources.
[2] A complete k-partite graph is a k-partite graph in which there is an edge between every pair of vertices from different independent sets.
These graphs are described by notation with a capital letter K subscripted by a sequence of the sizes of each set in the partition.
For instance, K2,2,2 is the complete tripartite graph of a regular octahedron, which can be partitioned into three independent sets each consisting of two opposite vertices.