In graph theory, a part of mathematics, a k-partite graph is a graph whose vertices are (or can be) partitioned into k different independent sets. Equivalently, it is a graph that can be colored with k colors, so that no two endpoints of an edge have the same color. When k = 2 these are the bipartite graphs, and when k = 3 they are called the tripartite graphs.

Bipartite graphs may be recognized in polynomial time but, for any k > 2 it is NP-complete, given an uncolored graph, to test whether it is k-partite.[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]

Example complete k-partite graphs
K2,2,2
Graph of octahedron
K3,3,3
Graph of complex generalized octahedron
K2,2,2,2
Graph of 16-cell

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. A complete multipartite graph is a graph that is complete k-partite for some k.[3] The Turán graphs are the special case of complete multipartite graphs in which the independent sets differ in size by at most one vertex. Complete k-partite graphs, complete multipartite graphs, and their complement graphs, the cluster graphs, are special cases of cographs, and can be recognized in polynomial time even when the partition is not supplied as part of the input.

References

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  1. ^ Garey, M. R.; Johnson, D. S. (1979), Computers and Intractability: A Guide to the Theory of NP-Completeness, W.H. Freeman, GT4, ISBN 0-7167-1045-5.
  2. ^ Hotho, Andreas; Jäschke, Robert; Schmitz, Christoph; Stumme, Gerd (2006), "FolkRank : A Ranking Algorithm for Folksonomies", LWA 2006: Lernen - Wissensentdeckung - Adaptivität, Hildesheim, October 9th-11th 2006, pp. 111–114.
  3. ^ Chartrand, Gary; Zhang, Ping (2008), Chromatic Graph Theory, CRC Press, p. 41, ISBN 9781584888017.

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Glossary of graph theory

sets of vertices in a tree decomposition. balanced A bipartite or multipartite graph is balanced if each two subsets of its vertex partition have sizes

Complete bipartite graph

matching from a complete bipartite graph Complete multipartite graph, a generalization of complete bipartite graphs to more than two sets of vertices Biclique

Directed graph

complete graph with the edges replaced by pairs of inverse arcs). It follows that a complete digraph is symmetric. Semicomplete multipartite digraphs

1-planar graph

many graphs. A complete classification of the 1-planar complete graphs, complete bipartite graphs, and more generally complete multipartite graphs is known

Turán graph

The Turán graph, denoted by T ( n , r ) {\displaystyle T(n,r)} , is a complete multipartite graph; it is formed by partitioning a set of n {\displaystyle

Bipartite graph

bipartite graph whose vertices can be ordered so that the vertex neighborhoods are contiguous Multipartite graph, a generalization of bipartite graphs to more

Distance-regular graph

{\displaystyle G,} unless G {\displaystyle G} is a cycle graph or a complete multipartite graph. If G {\displaystyle G} is strongly regular, then n ≤ 4

Graph coloring

signed graphs and gain graphs. Critical graph Graph coloring game Graph homomorphism Hajós construction Mathematics of Sudoku Multipartite graph Uniquely