2025/06/27 by Monks, Rebecca, Nixon, Anthony · 1 citation
#05C10 #52C25 #Combinatorics (math.CO) #FOS: Mathematics
paper · doi:10.48550/arxiv.2506.22214
A bar-joint framework (G,p) in Euclidean d-space is rigid if the only edge-length-preserving continuous motions arise from isometries of ℝd. In the generic case, rigidity is determined by the generic d-dimensional rigidity matroid of G. The combinatorial nature of this matroid is well understood when d=1,2 but open when d≥ 3. Jackson and Jordán 2005 characterised independence in this matroid for connected graphs with minimum degree at most d+1 and maximum degree at most d+2. Their characterisation is known to be false for (d+2)-regular graphs when d≥ 4 but when d=3 it remained open. Indeed they conjectured that their characterisation extends to 5-regular graphs when d=3. The purpose of this article is to prove their conjecture. That is, we prove that every 5-regular graph that has at most 3n-6 edges in any subgraph on n≥ 3 vertices is independent in the generic 3-dimensional rigidity matroid.