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	~~Hello! My name~~ is ~~Ilana~~. <br>It is a ~~little about myself: I live in France, my city~~ of ~~Alfortville~~. <br>~~It's called often Eastern or cultural capital~~ of ~~ILE~~-DE-~~FRANCE~~. I'~~ve married 4 years ago~~.<br>I have ~~two children -~~ a ~~son~~ (~~Felisha~~) and the ~~daughter~~ (~~Raymundo~~). ~~We all like Bowling~~.<br><br>~~Feel free~~ to ~~surf to my website :~~: [~~http:~~//~~yscon~~.~~realmind~~.~~net~~/xe/~~?document_srl~~=~~304504 women omega watches~~]		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]] <math>U{}^r_n</math>, the circuits are the sets of exactly <math>r+1</math> elements. Therefore, a uniform matroid is Eulerian if and only if <math>r+1</math> is a divisor of <math>n</math>. For instance, the <math>n</math>-point lines <math>U{}^2_n</math> are Eulerian if and only if <math>n</math> 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 [[complement (set theory)\|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 {{harvtxt\|Welsh\|1969}} as a generalization of the [[Eulerian graph]]s, 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 matroid]]s of Eulerian graphs are examples of Eulerian matroids.<ref name="w69">{{citation
			\| last = Welsh \| first = D. J. A. \| authorlink = Dominic Welsh
			\| journal = [[Journal of Combinatorial Theory]]
			\| mr = 0237368
			\| pages = 375–377
			\| title = Euler and bipartite matroids
			\| volume = 6
			\| year = 1969}}.</ref>

			The definition of an Eulerian graph above allows graphs that are disconnected, so not every such graph has an Euler tour. {{harvtxt\|Wilde\|1975}} observes that the graphs that have Euler tours can be characterized in an alternative way that generalizes to matroids: a graph <math>G</math> has an Euler tour if and only if it can be formed from some other graph <math>H</math>, and a cycle <math>C</math> in <math>H</math>, by [[graph minor\|contracting]] the edges of <math>H</math> that do not belong to <math>C</math>. In the contracted graph, <math>C</math> 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 [[matroid minor\|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 matroid]]s, matroids [[Matroid representation\|representable]] over [[GF(2)]]:
			a binary matroid is Eulerian if and only if it is the contraction of another binary matroid onto a circuit.<ref>{{citation
			\| last = Wilde \| first = P. J.
			\| journal = [[Journal of Combinatorial Theory]]
			\| mr = 0384577
			\| pages = 260–264
			\| series = Series B
			\| title = The Euler circuit theorem for binary matroids
			\| volume = 18
			\| year = 1975}}.</ref>

			==Duality with bipartite matroids==
			For [[planar graph]]s, the properties of being Eulerian and [[bipartite graph\|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.<ref name="w69"/><ref>{{citation
			\| last1 = Harary \| first1 = Frank \| author1-link = Frank Harary
			\| last2 = Welsh \| first2 = Dominic \| author2-link = Dominic Welsh
			\| contribution = Matroids versus graphs
			\| doi = 10.1007/BFb0060114
			\| location = Berlin
			\| mr = 0263666
			\| pages = 155–170
			\| publisher = Springer
			\| series = Lecture Notes in Mathematics
			\| title = The Many Facets of Graph Theory (Proc. Conf., Western Mich. Univ., Kalamazoo, Mich., 1968)
			\| volume = 110
			\| year = 1969}}.</ref>

			For matroids that are not binary, the duality between Eulerian and bipartite matroids may break down. For instance, the uniform matroid <math>U{}^2_6</math> is Eulerian but its dual <math>U{}^4_6</math> is not bipartite, as its circuits have size five. The self-dual uniform matroid <math>U{}^3_6</math> 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 {{harvtxt\|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.<ref>{{citation
			\| last = Shikare \| first = M. M.
			\| issue = 2
			\| journal = Indian Journal of Pure and Applied Mathematics
			\| mr = 1820861
			\| pages = 215–219
			\| title = New characterizations of Eulerian and bipartite binary matroids
			\| url = http://www.dli.gov.in/rawdataupload/upload/insa/INSA_1/20005b01_215.pdf
			\| volume = 32
			\| year = 2001}}.</ref> {{harvtxt\|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.<ref>{{citation
			\| last1 = Lovász \| first1 = László \| author1-link = László Lovász
			\| last2 = Seress \| first2 = Ákos
			\| doi = 10.1006/eujc.1993.1027
			\| issue = 3
			\| journal = European Journal of Combinatorics
			\| mr = 1215334
			\| pages = 241–250
			\| title = The cocycle lattice of binary matroids
			\| volume = 14
			\| year = 1993}}.</ref>

			==Computational complexity==
			Any algorithm that tests whether a given matroid is Eulerian, given access to the matroid via an [[matroid oracle\|independence oracle]], must perform an exponential number of oracle queries, and therefore cannot take polynomial time.<ref>{{citation
			\| last1 = Jensen \| first1 = Per M.
			\| last2 = Korte \| first2 = Bernhard
			\| doi = 10.1137/0211014
			\| issue = 1
			\| journal = [[SIAM Journal on Computing]]
			\| mr = 646772
			\| pages = 184–190
			\| title = Complexity of matroid property algorithms
			\| volume = 11
			\| year = 1982}}.</ref>

			==Eulerian extension==
			If <math>M</math> is a binary matroid that is not Eulerian, then it has a unique '''Eulerian extension''', a binary matroid <math>\bar M</math> whose elements are the elements of <math>M</math> together with one additional element <math>e</math>, such that the restriction of <math>\bar M</math> to the elements of <math>M</math> is isomorphic to <math>M</math>. The dual of <math>\bar M</math> is a bipartite matroid formed from the dual of <math>M</math> by adding <math>e</math> to every odd circuit.<ref>{{citation
			\| last = Sebő \| first = András
			\| contribution = The cographic multiflow problem: an epilogue
			\| location = Providence, RI
			\| mr = 1105128
			\| pages = 203–234
			\| publisher = Amer. Math. Soc.
			\| series = DIMACS Ser. Discrete Math. Theoret. Comput. Sci.
			\| title = Polyhedral combinatorics (Morristown, NJ, 1989)
			\| volume = 1
			\| year = 1990}}.</ref>

			==References==
			{{reflist}}

			[[Category:Matroid theory]]

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