Bipartite matroid
In mathematics, a bipartite matroid is a matroid all of whose circuits have even size.
Example
A uniform matroid is bipartite if and only if is an odd number, because the circuits in such a matroid have size.Relation to bipartite graphs
Bipartite matroids were defined by as a generalization of the bipartite graphs, graphs in which every cycle has even size. A graphic matroid is bipartite if and only if it comes from a bipartite graph.Duality with Eulerian matroids
An Eulerian graph is one in which all vertices have even degree; Eulerian graphs may be disconnected. For planar graphs, the properties of being bipartite and Eulerian are dual: a planar graph is bipartite if and only if its dual graph is Eulerian. As Welsh showed, this duality extends to binary matroids: a binary matroid is bipartite if and only if its dual matroid is an Eulerian matroid, a matroid that can be partitioned into disjoint circuits.For matroids that are not binary, the duality between Eulerian and bipartite matroids may break down. For instance, the uniform matroid is non-bipartite but its dual is Eulerian, as it can be partitioned into two 3-cycles. The self-dual uniform matroid is bipartite but not Eulerian.