The Fano plane has two kinds of circuits: sets of three collinear points, and sets of four points that do not contain any line.
Thus, the Fano plane is also Eulerian.
Eulerian matroids were defined by Welsh (1969) as a generalization of the Eulerian graphs, graphs in which every vertex has even degree.
By Veblen's theorem the edges of every such graph may be partitioned into simple cycles, from which it follows that the graphic matroids of Eulerian graphs are examples of Eulerian matroids.
Wilde (1975) observes that the graphs that have Euler tours can be characterized in an alternative way that generalizes to matroids: a graph
has an Euler tour if and only if it can be formed from some other graph
generally stops being a simple cycle and becomes instead an Euler tour.
Analogously, Wilde considers the matroids that can be formed from a larger matroid by contracting the elements that do not belong to some particular circuit.
He shows that this property is trivial for general matroids (it implies only that each element belongs to at least one circuit) but can be used to characterize the Eulerian matroids among the binary matroids, matroids representable over GF(2): a binary matroid is Eulerian if and only if it is the contraction of another binary matroid onto a circuit.
[2] For planar graphs, the properties of being Eulerian and bipartite are dual: a planar graph is Eulerian if and only if its dual graph is bipartite.
For instance, the uniform matroid
is not bipartite, as its circuits have size five.
The self-dual uniform matroid
Because of the correspondence between Eulerian and bipartite matroids among the binary matroids, the binary matroids that are Eulerian may be characterized in alternative ways.
The characterization of Wilde (1975) is one example; two more are that a binary matroid is Eulerian if and only if every element belongs to an odd number of circuits, if and only if the whole matroid has an odd number of partitions into circuits.
[4] Lovász & Seress (1993) collect several additional characterizations of Eulerian binary matroids, from which they derive a polynomial time algorithm for testing whether a binary matroid is Eulerian.
[5] Any algorithm that tests whether a given matroid is Eulerian, given access to the matroid via an independence oracle, must perform an exponential number of oracle queries, and therefore cannot take polynomial time.
In particular, it is difficult to distinguish a uniform matroid on a set of
elements, with all cycles of size
, from a paving matroid that differs from the uniform matroid in having two complementary cycles of size
Any oracle algorithm, applied to the uniform matroid, must make
queries, an exponential number, to verify that the input is not instead an instance of the paving matroid.
is a binary matroid that is not Eulerian, then it has a unique Eulerian extension, a binary matroid
is a bipartite matroid formed from the dual of