2021/10/25 by James Oxley, Zach Walsh, Oxley, James +1
Computer Science · #05B35 #Advanced Graph Theory Research #Combinatorics (math.CO) #Complexity and Algorithms in Graphs #FOS: Mathematics
paper · pdf · doi:10.48550/arxiv.2110.13120
openalex publication_date 2021/10/25 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Halin proved that every minimally k-connected graph has a vertex of degree k. More generally, does every minimally vertically k-connected matroid have a k-element cocircuit? Results of Murty and Wong give an affirmative answer when k ≤ 3. We show that every minimally vertically 4-connected matroid with at least six elements has a 4-element cocircuit, or a 5-element cocircuit that contains a triangle, with the exception of a specific non-binary 9-element matroid. Consequently, every minimally vertically 4-connected binary matroid with at least six elements has a 4-element cocircuit.