Combinatorial Representations

Cameron, Peter J.; Gadouleau, Maximilien; Riis, Søren

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.

Citation

Cameron, P. J., Gadouleau, M., & Riis, S. (2013). Combinatorial Representations. Journal of Combinatorial Theory, Series A, 120(3), 671-682. https://doi.org/10.1016/j.jcta.2012.12.002

• Journal Article Type: Article
• Publication Date: Apr 1, 2013
• Deposit Date: Apr 10, 2013
• Publicly Available Date: Oct 19, 2015
• Journal: Journal of Combinatorial Theory, Series A
• Print ISSN: 0097-3165
• Electronic ISSN: 1096-0899
• Publisher: Elsevier
• Peer Reviewed: Peer Reviewed
• Volume: 120
• Issue: 3
• Pages: 671-682
• DOI: https://doi.org/10.1016/j.jcta.2012.12.002
• Public URL: https://durham-repository.worktribe.com/output/1456395