2021/06/01 by Cameron Crenshaw, Crenshaw, Cameron, James Oxley +1
Computer Science · Engineering · #05B35 #Advanced Graph Theory Research #Combinatorics (math.CO) #FOS: Mathematics #Interconnection Networks and Systems #graph theory and CDMA systems
paper · pdf · doi:10.48550/arxiv.2106.00852
openalex publication_date 2021/06/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
The cogirth, g^∗(M), of a matroid M is the size of a smallest cocircuit of M. Finding the cogirth of a graphic matroid can be done in polynomial time, but Vardy showed in 1997 that it is NP-hard to find the cogirth of a binary matroid. In this paper, we show that g^∗(M)≤ (1)/(2)\vert E(M)\vert when M is binary, unless M simplifies to a projective geometry. We also show that, when equality holds, M simplifies to a Bose-Burton geometry, that is, a matroid of the form PG(r-1,2)-PG(k-1,2). These results extend to matroids representable over arbitrary finite fields.