Combinatorial representations

Authors:
Peter J. Cameron;Maximilien Gadouleau;SøRen Riis
Affiliations:
School of Mathematical Sciences, Queen Mary, University of London, Mile End Road, London E1 4NS, UK;School of Engineering and Computing Sciences, Durham University, South Road, Durham DH1 3LE, UK;School of Electronic Engineering and Computer Science, Queen Mary, University of London, Mile End Road, London E1 4NS, UK
Venue:
Journal of Combinatorial Theory Series A
Year:
2013

Citing 3
Cited 0

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A non-Shannon-type conditional inequality of information quantities

IEEE Transactions on Information Theory

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Abstract

This paper introduces combinatorial representations, which generalise the notion of linear representations of matroids. We show that any family of subsets of the same cardinality has a combinatorial representation via matrices. We then prove that any graph is representable over all alphabets of size larger than some number depending on the graph. We also provide a characterisation of families representable over a given alphabet. Then, we associate a rank function and a closure operator to any representation which help us determine some criteria for the functions used in a representation. While linearly representable matroids can be viewed as having representations via matrices with only one row, we conclude this paper by an investigation of representations via matrices with only two rows.