Graph, bipartite

2020 Mathematics Subject Classification: Primary: 05C [MSN][ZBL]

bichromatic graph

A graph whose set $V$ of vertices can be partitioned into two disjoint sets $V'$ and $V''$ (i.e. $V=V'\cup V''$, $V'\cap V''=\emptyset$) so that each edge connects some vertex of $V'$ with some vertex of $V''$. A graph is bipartite if and only if all its simple cycles have even length. Another frequently used definition of a bipartite graph is a graph in which two subsets $V'$ and $V''$ of vertices (parts) are given in advance. Bipartite graphs are convenient for the representation of binary relations between elements of two different types — e.g. the elements of a given set and a subset of it yield the relation of "membership of an element to a subset", for executors and types of jobs one has the relation "a given executor can carry out a given job", etc.

An important problem concerning bipartite graphs is the study of matchings, that is, families of pairwise non-adjacent edges. Such problems occur, for example, in the theory of scheduling (partitioning of the edges of a bipartite graph into a minimal number of disjoint matchings), in the problem of assignment (finding the maximum number of elements in a matching), etc. The cardinality of the maximum matching in a bipartite graph is

$$|V'|-\max_{A'\subseteq V'}(|A'|-|V''(A')|),$$

where $V''(A')$ is the number of vertices of $V''$ adjacent to at least one vertex of $A'$. A complete bipartite graph is a bipartite graph in which any two vertices belonging to different subsets are connected by an edge (e.g. the graph $K_{3,3}$, see Graph, planar, Figure 1).

A generalization of the concept of a bipartite graph is the concept of a multipartite or $k$-partite graph, i.e. a graph in which the vertices are partitioned into $k$ subsets so that each edge connects vertices belonging to different subsets.

References

[1]	O. Ore, "Theory of graphs" , Amer. Math. Soc. (1962)

Comments

The complete bipartite graph on vertex sets of size $a,b$ is denoted $K_{a,b}$. Similarly the complete multipartite graph on $k$ vertex sets of sizes $a_1,\ldots,a_k$ is denoted $K_{a_1,\ldots,a_k}$. The complement of a complete multipartite $K_{a_1,\ldots,a_k}$ is a disjoint union of complete graphs $K_{a_1} \sqcup \cdots \sqcup K_{a_k}$.

References

[a1]	F. Harary, "Graph theory" , Addison-Wesley (1969) pp. Chapt. 9
[a2]	R.J. Wilson, "Introduction to graph theory" , Longman (1985)
[b1]	Biggs, Norman Algebraic graph theory 2nd ed. Cambridge University Press (1994) ISBN 0-521-45897-8 Zbl 0797.05032