All 880 Complex Realizations of the Maximal 10-Vertex Laman Graph Can Be Real
Year: 2026
Type: preprint
Abstract: All 880 complex realizations of the maximal 10-vertex Laman graph L880 can be simultaneously real. This record contains a computer-assisted proof, pending independent verification, that r2(L880) = r2(10) = c2(10) = 880, which settles φ10 = 1, the smallest case of Open Problem 5 in Grasegger, Koutschan and Tsigaridas (Experimental Mathematics, 2020; Lower bounds on the number of realizations of rigid graphs). The sharpest earlier result, by Bartzos, Emiris, Legerský and Tsigaridas, certified 860 real realizations and observed 868 numerically. The proof fixes 17 explicit positive rational bar lengths, so that the pinned base triangle lies in Q^2 with no square roots. Exact rational Krawczyk interval arithmetic, with no floating-point operations, certifies 440 pairwise-disjoint contractive boxes in R^14, and reflection across the base triangle doubles them to 880 distinct nonsingular real realizations modulo SE(2). A standalone implementation of Theorem 4.7 of Capco et al. confirms c2(L880) = 880. Gluing copies of this framework along the rational base triangle (the fan construction) raises the certified lower bound on liminf r2(n)^(1/n) from 430^(1/7) ≈ 2.377989 to 440^(1/7) ≈ 2.385811. At Bartzos et al.'s unmodified bar lengths the same machinery certifies the lower bounds r2(L880) ≥ 868 and r3(G160) ≥ 132 for the 8-vertex maximal Geiringer graph in R^3. The deposit contains the paper, the three certificates, the pure-stdlib Python verifiers and tests, and SHA256 checksums. The record is licensed CC BY 4.0. The scripts and certificates are Apache-2.0 under the repository LICENSE. Certificates for Prism_6, L136 and L344 are not included, and the glued bounds for n = 11 to 13 listed in the repository README depend on certificates that are not yet committed. Researched & verified by models across Gemini, Claude, Grok, OpenAI, and Muse families
Author: Arifuddin, Ahraz
Language: English
Subfield: Computational Theory and Mathematics
Field: Computer Science
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7225782615