2018/04/19 by Nick Brettell, Rutger Campbell, Deborah Chun +2 · 3 citations
Computer Science · Mathematics · #Advanced Graph Theory Research #Complexity and Algorithms in Graphs #Generalization #Graphic matroid #Integer (computer science) #Interconnection Networks and Systems #Matroid #Matroid partitioning #Partition (number theory) #Property (philosophy) #math.CO #msc:05B35
paper · pdf · open access · doi:10.1137/18m1182255
published in SIAM Journal on Discrete Mathematics 33(1), 358-372 (Society for Industrial and Applied Mathematics) · 18 pages
arxiv created 2018/04/19 · openalex created_date 2018/04/24 · openalex publication_date 2019/01/01 · arxiv updated 2020/06/02 · openalex updated_date 2026/08/05
We consider matroids with the property that every subset of the ground set of size t is contained in both an ℓ-element circuit and an ℓ-element cocircuit; we say that such a matroid has the (t,ℓ)-property. We show that for any positive integer t, there is a finite number of matroids with the (t,ℓ)-property for ℓ<2t; however, matroids with the (t,2t)-property form an infinite family. We say a matroid is a t-spike if there is a partition of the ground set into pairs such that the union of any t pairs is a circuit and a cocircuit. Our main result is that if a sufficiently large matroid has the (t,2t)-property, then it is a t-spike. Finally, we present some properties of t-spikes.