A bipartite graph, also called a bigraph, is a graph whose vertex set can be decomposed into two disjoint
sets such that no two graph vertices within the
same set are adjacent. The two vertex classes are
said to form a bipartition. A bipartite graph is
a special case of a k-partite graph with
. The illustration above shows some
bipartite graphs, with vertices in each graph
colored based on to which part of the bipartition
they belong.
Bipartite graphs are equivalent to two-colorable graphs. All acyclic graphs are bipartite. A cyclic graph is bipartite iff all its cycles are of even length (Skiena 1990, p. 213).
If a bipartite graph has bipartition classes 
and 
, then any Hamiltonian cycle
alternates between 
and 
and so requires 
.
Similarly, any Hamiltonian path alternates between
the classes, and so requires 
. Thus a connected
bipartite graph with unequal bipartition sizes is
nonhamiltonian and one with bipartition
sizes differing by more than 1 is untraceable.
These converses do not hold in general: the E graph and
H graph have equal bipartition
sizes but are nonhamiltonian (indeed untraceable),
while the fork graph and spoke
graph 
have bipartition sizes differing by 1 but are also
untraceable.
Families of bipartite graphs include
1. acyclic graphs (i.e., trees and forests),
2. book graphs 
,
4. crown graphs 
,
5. cycle graphs 
,
6. gear graphs,
7. grid graphs,
8. Haar graphs,
9. Hadamard graphs,
10. hypercube graphs 
,
11. knight graphs,
12. ladder graphs,
13. ladder rung graphs 
(which are forests).
14. path graphs 
(which are trees),
17. stacked book graphs,
18. star graphs 
(which are trees).
König's line coloring theorem states that every bipartite graph is a class 1 graph. The König-Egerváry theorem states that the matching number (i.e., size of a maximum independent edge set) equals the vertex cover number (i.e., size of a minimum vertex cover) for a bipartite graph.
A graph may be tested in the Wolfram Language to see if it is a bipartite graph using BipartiteGraphQ[g].
The numbers of bipartite graphs on 
, 2, ... nodes are 1, 2,
3, 7, 13, 35, 88, 303, ... (OEIS A033995).
The numbers of connected bipartite graphs on 
, 2 ... nodes
are 1, 1, 1, 3, 5, 17, 44, 182, ... (OEIS A005142).
