Transversal (combinatorics): Difference between revisions

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{{Short description|Set that intersects every one of a family of sets}}

{{Other uses|Transversal (disambiguation)}}

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* One variation is that there is a [[bijection]] ''f'' from the transversal to ''C'' such that ''x'' is an element of ''f''(''x'') for each ''x'' in the transversal. In this case, the transversal is also called a '''system of distinct representatives''' (SDR).<ref name="lp">{{Cite Lovasz Plummer}}</ref>{{rp|29}}

* The other, less commonly used, does not require a one-to-one relation between the elements of the transversal and the sets of ''C''. In this situation, the members of the '''system of representatives''' are not necessarily distinct.<ref>{{harvnbcitation|last1=Roberts|Tesmanfirst1=Fred S.|title=Applied Combinatorics|year=2009|locedition=pg.2nd|place=Boca 692Raton|publisher=CRC Press|isbn=978-1-4200-9982-9|last2=Tesman|first2=Barry}}</ref>{{Rp|692}}<ref>{{harvnbcitation|last=Brualdi|first=Richard A.|title=Introductory Combinatorics|year=2010|locedition=pg.5th|place=Upper 322Saddle River, NJ|publisher=Prentice Hall|isbn=978-0-13-602040-0}}</ref>{{Rp|322}}

In [[computer science]], computing transversals is useful in several application domains, with the input [[family of sets]] often being described as a [[hypergraph]].

In [[set theory]], the [[axiom of choice]] is equivalent to the statement that every [[partition of a set|partition]] has a transversal.<ref>{{cite web|url=https://plato.stanford.edu/entries/axiom-choice/|title=The Axiom of Choice|website=Stanford Encyclopedia of Philosophy|first=Bell|last=John|date=December 10, 2021|access-date=December 2, 2024|quote=Let us call Zermelo’s 1908 formulation the combinatorial axiom of choice: CAC: Any collection of mutually disjoint nonempty sets has a transversal.}}</ref>

==Existence and number==

A fundamental question in the study of SDR is whether or not an SDR exists. [[Hall's marriage theorem]] gives necessary and sufficient conditions for a finite collection of sets, some possibly overlapping, to have a transversal. The condition is that, for every integer ''k'', every collection of ''k'' sets must contain in common at least ''k'' different elements.<ref name="lp" />{{rp|29}}

The following refinement by [[H. J. Ryser]] gives lower bounds on the number of such SDRs.<ref>{{harvnbcitation|last=Ryser|first=Herbert John|title=Combinatorial Mathematics|year=1963|locseries=pThe Carus Mathematical Monographs #14|publisher=Mathematical Association of America|author-link=H. 48J. Ryser}}</ref>{{Rp|48}}

''Theorem''. Let ''S''₁, ''S''₂, ..., ''S''_''m'' be a collection of sets such that <math>S_{i_1} \cup S_{i_2} \cup \dots \cup S_{i_k}</math> contains at least ''k'' elements for ''k'' = 1,2,...,''m'' and for all ''k''-combinations {<math>i_1, i_2, \ldots, i_k</math>} of the integers 1,2,...,''m'' and suppose that each of these sets contains at least ''t'' elements. If ''t'' ≤ ''m'' then the collection has at least ''t'' ! SDRs, and if ''t'' > ''m'' then the collection has at least ''t'' ! / (''t'' - ''m'')! SDRs.

== Relation to matching and covering ==

One can construct a [[bipartite graph]] in which the vertices on one side are the sets, the vertices on the other side are the elements, and the edges connect a set to the elements it contains. Then, a transversal (defined as a system of ''distinct'' representatives) is equivalent to a [[Perfect matching|'''[[perfect matching]]''']] in this graph.

One can construct a [[hypergraph]] in which the vertices are the elements, and the hyperedges are the sets. Then, a transversal is(defined equivalentas toa system of ''not-necessarily-distinct'' representatives) is a [[Vertex cover in hypergraphs|'''vertex cover''' in a hypergraph]].

==Examples==

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== Common Transversalstransversals ==

A '''common transversal''' of the collections ''A'' and ''B'' (where <math>|A| = |B| = n</math>) is a set that is a transversal of both ''A'' and ''B''. The collections ''A'' and ''B'' have a common transversal if and only if, for all <math>I, J \subset \{1,...,n\}</math>,

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A '''partial transversal''' is a set containing at most one element from each member of the collection, or (in the stricter form of the concept) a set with an injection from the set to ''C''. The transversals of a finite collection ''C'' of finite sets form the basis sets of a [[matroid]], the '''transversal matroid''' of ''C''. The independent sets of the transversal matroid are the partial transversals of ''C''.<ref>{{citation|last=Oxley|first=James G.|title=Matroid Theory|volume=3|page=48|year=2006|series=Oxford graduate texts in mathematics|publisher=Oxford University Press|isbn=978-0-19-920250-8|authorlink=James Oxley}}.</ref>

An '''independent transversal''' (also called a [[Rainbow independent set|'''[[rainbow -independent set]]''']] or '''independent system of representatives''') is a transversal which is also an [[Independent set (graph theory)|independent set]] of a given graph. To explain the difference in figurative terms, consider a faculty with ''m'' departments, where the faculty dean wants to construct a committee of ''m'' members, one member per department. Such a committee is a transversal. But now, suppose that some faculty members dislike each other and do not agree to sit in the committee together. In this case, the committee must be an independent transversal, where the underlying graph describes the "dislike" relations.<ref>{{Cite journal|last=Haxell|first=P.|date=2011-11-01|title=On Forming Committees|url=https://www.tandfonline.com/doi/abs/10.4169/amer.math.monthly.118.09.777|journal=The American Mathematical Monthly|volume=118|issue=9|pages=777–788|doi=10.4169/amer.math.monthly.118.09.777|s2cid=27202372 |issn=0002-9890|url-access=subscription}}</ref>

Another generalization of the concept of a transversal would be a set that just has a non-empty intersection with each member of ''C''. An example of the latter would be a [[Bernstein set|'''[[Bernstein set]]''']], which is defined as a set that has a non-empty intersection with each set of ''C'', but contains no set of ''C'', where ''C'' is the collection of all [[perfect setsset]]s of a topological [[Polish space]]. As another example, let ''C'' consist of all the lines of a [[projective plane]], then a [[blocking set]] in this plane is a set of points which intersects each line but contains no line.

==Category theory==

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*[[Permanent (mathematics)|Permanent]]

==NotesReferences==

{{reflist}}

==ReferencesFurther reading==

* {{citation|last=Brualdi|first=Richard A.|title=Introductory Combinatorics|edition=5th|year=2010|publisher=Prentice Hall|place=Upper Saddle River, NJ |isbn=0-13-602040-2}}

* [[Eugene Lawler|Lawler, E. L.]] Combinatorial Optimization: Networks and Matroids. 1976.

* [[Leon Mirsky|Mirsky, Leon]] (1971). ''Transversal Theory: An account of some aspects of combinatorial mathematics.'' Academic Press. {{ISBN|0-12-498550-5}}.

* {{citation|last1=Roberts|first1=Fred S.|last2=Tesman|first2=Barry|title=Applied Combinatorics|edition=2nd|publisher=CRC Press|place=Boca Raton|year=2009|isbn=978-1-4200-9982-9}}

* {{citation|last=Ryser|first=Herbert John|author-link=H. J. Ryser|title=Combinatorial Mathematics|series=The Carus Mathematical Monographs #14|year=1963|publisher=Mathematical Association of America}}

[[Category:Combinatorics]]

[[Category:Group theory]]

[[Category:SetFamilies familiesof sets]]