Independence system partitionable into circuits
In matroid theory, an Eulerian matroid is a matroid whose elements can be partitioned into a collection of disjoint circuits.
Examples.
In a uniform matroid formula_1, the circuits are the sets of exactly formula_2 elements. Therefore, a uniform matroid is Eulerian if and only if formula_2 is a divisor of formula_4. For instance, the formula_4-point lines formula_6 are Eulerian if and only if formula_4 is divisible by three.
The Fano plane has two kinds of circuits: sets of three collinear points, and sets of four points that do not contain any line. The three-point circuits are the complements of the four-point circuits, so it is possible to partition the seven points of the plane into two circuits, one of each kind. Thus, the Fano plane is also Eulerian.
Relation to Eulerian graphs.
Eulerian matroids were defined by 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.
The definition of an Eulerian graph above allows graphs that are disconnected, so not every such graph has an Euler tour. observes that the graphs that have Euler tours can be characterized in an alternative way that generalizes to matroids: a graph formula_8 has an Euler tour if and only if it can be formed from some other graph formula_9, and a cycle formula_10 in formula_9, by contracting the edges of formula_9 that do not belong to formula_10. In the contracted graph, formula_10 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.
Duality with bipartite matroids.
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. As Welsh showed, this duality extends to binary matroids: a binary matroid is Eulerian if and only if its dual matroid is a bipartite matroid, a matroid in which every circuit has even cardinality.
For matroids that are not binary, the duality between Eulerian and bipartite matroids may break down. For instance, the uniform matroid formula_15 is Eulerian but its dual formula_16 is not bipartite, as its circuits have size five. The self-dual uniform matroid formula_17 is bipartite but not Eulerian.
Alternative characterizations.
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 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. collect several additional characterizations of Eulerian binary matroids, from which they derive a polynomial time algorithm for testing whether a binary matroid is Eulerian.
Computational complexity.
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 formula_18 elements, with all cycles of size formula_19, from a paving matroid that differs from the uniform matroid in having two complementary cycles of size formula_4. The paving matroid is Eulerian but the uniform matroid is not. Any oracle algorithm, applied to the uniform matroid, must make formula_21 queries, an exponential number, to verify that the input is not instead an instance of the paving matroid.
Eulerian extension.
If formula_22 is a binary matroid that is not Eulerian, then it has a unique Eulerian extension, a binary matroid formula_23 whose elements are the elements of formula_22 together with one additional element formula_25, such that the restriction of formula_23 to the elements of formula_22 is isomorphic to formula_22. The dual of formula_23 is a bipartite matroid formed from the dual of formula_22 by adding formula_25 to every odd circuit.
References.
 ;templatestyles src="Reflist/styles.css" / ;