Partition matroid

File:Partition matroid.svg

For the bipartite graph shown on the left, each vertex in the first column is assigned a unique color. Then, each edge is colored based on which colored vertex it is connected to. Each independent set in the matroid on the right is allowed a maximum of 1 edge of each color. Thus, the matroid is a partition matroid with $|C_i|=3$ and $d_i=1$ for all $i$. The latter condition means this matroid is also a transversal matroid.

In mathematics, a partition matroid or partitional matroid is a matroid that is a direct sum of uniform matroids. It is defined over a base set in which the elements are partitioned into different categories. For each category, there is a capacity constraint - a maximum number of allowed elements from this category. The independent sets of a partition matroid are exactly the sets in which, for each category, the number of elements from this category is at most the category capacity.

Formal definition

Let $C_i$ be a collection of disjoint sets ("categories"). Let $d_i$ be integers with $0\le d_i\le |C_i|$ ("capacities"). Define a subset $I\subseteq \bigcup _i{C}_i$ to be "independent" when, for every index $i$, $|I\cap C_i|\le d_i$. The sets satisfying this condition form the independent sets of a matroid, called a partition matroid.

The sets $C_i$ are called the categories or the blocks of the partition matroid.

A basis of the partition matroid is a set whose intersection with every block $C_i$ has size exactly $d_i$. A circuit of the matroid is a subset of a single block $C_i$ with size exactly $d_i+1$. The rank of the matroid is $\sum d_i$.

Every uniform matroid $U{}_n^r$ is a partition matroid, with a single block $C_1$ of $n$ elements and with $d_1=r$. 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 $d_i=1$. The partitions that obey this more restrictive definition are the transversal matroids of the family of disjoint sets given by their blocks.

Properties

As with the uniform matroids they are formed from, the dual matroid of a partition matroid is also a partition matroid, and every 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 $(U,V)$, the sets of edges satisfying the condition that no two edges share an endpoint in $U$ are the independent sets of a partition matroid with one block per vertex in $U$ and with each of the numbers $d_i$ equal to one. The sets of edges satisfying the condition that no two edges share an endpoint in $V$ 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.

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.

Clique complexes

A clique complex is a family of sets of vertices of a graph $G$ that induce complete subgraphs of $G$. A clique complex forms a matroid if and only if $G$ is a 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 intersections of families of partition matroids for which every $d_i=1$.

Enumeration

The number of distinct partition matroids that can be defined over a set of $n$ labeled elements, for $n=0,1,2,\dots$, is

1, 2, 5, 16, 62, 276, 1377, 7596, 45789, 298626, 2090910, ... (sequence A005387 in the OEIS).

The exponential generating function of this sequence is $f(x)=\exp (e^x(x-1)+2x+1)$.