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In mathematics, a '''partition matroid''' or '''partitional matroid''' is a [[matroid]] formed from a direct sum of [[uniform matroid]]s.<ref>{{citation
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| last = Recski | first = A.
| contribution = On partitional matroids with applications
| location = Amsterdam
| mr = 0389630
| pages = 1169–1179
| publisher = North-Holland
| series = Colloq. Math. Soc. Janós Bolyai
| title = Infinite and finite sets (Colloq., Keszthely, 1973; dedicated to P. Erdős on his 60th birthday), Vol. III
| volume = 10
| year = 1975}}.</ref>
 
==Definition==
Let <math>B_i</math> be a collection of [[disjoint sets]], and let <math>d_i</math> be integers with <math>0\le d_i\le |B_i|</math>. Define a set <math>I</math> to be "independent" when, for every index <math>i</math>, <math>|I\cap B_i|\le d_i</math>. Then the sets that are independent sets in this way form the independent sets of a [[matroid]], called a partition matroid. The sets <math>B_i</math> are called the blocks of the partition matroid. A basis of the matroid is a set whose intersection with every block <math>B_i</math> has size exactly <math>d_i</math>, and a circuit of the matroid is a subset of a single block <math>B_i</math> with size exactly <math>d_i+1</math>. The [[Matroid rank|rank]] of the matroid is <math>\sum d_i</math>.<ref>{{citation
| last = Lawler | first = Eugene L. | authorlink = Eugene Lawler
| location = Rinehart and Winston, New York
| mr = 0439106
| page = 272
| publisher = Holt
| title = Combinatorial Optimization: Networks and Matroids
| year = 1976}}.</ref>
 
Every [[uniform matroid]] <math>U{}^r_n</math> is a partition matroid, with a single block <math>B_1</math> of <math>n</math> elements and with <math>d_1=r</math>. Every partition matroid is the direct sum of a collection of uniform matroids, one for each of its blocks.
 
In some publications, the notion of a partition matroid is defined more restrictively, with every <math>d_i=1</math>. The partitions that obey this more restrictive definition are the [[transversal matroid]]s of the family of disjoint sets given by their blocks.<ref>E.g., see {{harvtxt|Kashiwabara|Okamoto|Uno|2007}}. {{harvtxt|Lawler|1976}} uses the broader definition but notes that the <math>d_i=1</math> restriction is useful in many applications.</ref>
 
==Properties==
As with the uniform matroids they are formed from, the [[dual matroid]] of a partition matroid is also a partition matroid, and every [[matroid minor|minor]] of a partition matroid is also a partition matroid. Direct sums of partition matroids are partition matroids as well.
 
==Matching==
A [[maximum matching]] in a graph is a set of edges that is as large as possible subject to the condition that no two edges share an endpoint. In a [[bipartite graph]] with bipartition <math>(U,V)</math>, the sets of edges satisfying the condition that no two edges share an endpoint in <math>U</math> are the independent sets of a partition matroid with one block per vertex in <math>U</math> and with each of the numbers <math>d_i</math> equal to one. The sets of edges satisfying the condition that no two edges share an endpoint in <math>V</math> are the independent sets of a second partition matroid. Therefore, the bipartite maximum matching problem can be represented as a [[matroid intersection]] of these two matroids.<ref>{{citation
| last1 = Papadimitriou | first1 = Christos H. | author1-link = Christos Papadimitriou
| last2 = Steiglitz | first2 = Kenneth | author2-link = Kenneth Steiglitz
| isbn = 0-13-152462-3
| location = Englewood Cliffs, N.J.
| mr = 663728
| pages = 289–290
| publisher = Prentice-Hall Inc.
| title = Combinatorial Optimization: Algorithms and Complexity
| year = 1982}}.</ref>
 
More generally the matchings of a graph may be represented as an intersection of two matroids if and only if every odd cycle in the graph is a triangle containing two or more degree-two vertices.<ref>{{citation
| last1 = Fekete | first1 = Sándor P.
| last2 = Firla | first2 = Robert T.
| last3 = Spille | first3 = Bianca
| arxiv = math/0212235
| doi = 10.1007/s001860300301
| issue = 2
| journal = Mathematical Methods of Operations Research
| mr = 2015015
| pages = 319–329
| title = Characterizing matchings as the intersection of matroids
| volume = 58
| year = 2003}}.</ref>
 
==Clique complexes==
A [[clique complex]] is a family of sets of vertices of a graph <math>G</math> that induce complete subgraphs of <math>G</math>. A clique complex forms a matroid if and only if <math>G</math> is a [[Glossary of graph theory|complete multipartite graph]], and in this case the resulting matroid is a partition matroid. The clique complexes are exactly the set systems that can be formed as [[Matroid intersection|intersections]] of families of partition matroids for which every <math>d_i=1</math>.<ref>{{citation
| last1 = Kashiwabara | first1 = Kenji
| last2 = Okamoto | first2 = Yoshio
| last3 = Uno | first3 = Takeaki
| doi = 10.1016/j.dam.2007.05.004
| issue = 15
| journal = Discrete Applied Mathematics
| mr = 2351976
| pages = 1910–1929
| title = Matroid representation of clique complexes
| volume = 155
| year = 2007}}. For the same results in a complementary form using independent sets in place of cliques, see {{citation
| last1 = Tyshkevich | first1 = R. I. | author1-link = Regina Tyshkevich
| last2 = Urbanovich | first2 = O. P.
| last3 = Zverovich | first3 = I. È.
| contribution = Matroidal decomposition of a graph
| location = Warsaw
| mr = 1097648
| pages = 195–205
| publisher = PWN
| series = Banach Center Publ.
| title = Combinatorics and graph theory (Warsaw, 1987)
| volume = 25
| year = 1989}}.</ref>
 
==Enumeration==
The number of distinct partition matroids that can be defined over a set of <math>n</math> labeled elements, for <math>n=0,1,2,\dots</math>, is
:1, 2, 5, 16, 62, 276, 1377, 7596, 45789, 298626, 2090910, ... {{OEIS|A005387}}.
The [[exponential generating function]] of this sequence is <math>f(x)=\exp(e^x(x-1)+2x+1)</math>.<ref>{{citation
| last = Recski | first = A.
| journal = Studia Scientiarum Mathematicarum Hungarica
| mr = 0379248
| pages = 247–249 (1975)
| title = Enumerating partitional matroids
| volume = 9
| year = 1974}}.</ref>
 
==References==
{{reflist}}
 
[[Category:Matroid theory]]
[[Category:Matching]]

Latest revision as of 02:54, 21 December 2014

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