{"id":"palomar/PALOMAR-2026-09-04-000002","url":"https://github.com/lennrt/palomar-formalizations/tree/debbfca8601d34a975a54d327f8dcba7ec6289df/zenodo-21925574/palomar","json":"https://data.palomar-registry.org/entries/PALOMAR-2026-09-04-000002-v1.json","repo":"https://github.com/lennrt/palomar-formalizations","tags":["math.CO","cs.CG","msc:05C10","msc:05C40","msc:05C83","msc:55N10","msc:57Q15","msc:68U05","graph-minors","tutte-embedding"],"score":0,"title":"Combinatorial core of the K₃,₃,₁-redundancy theorem for tetrahedral balls","trust":"high","domain":"math.CO","source":"palomar","authors":["Lennart Rudolph"],"reviews":0,"subject":"math.CO","summary":"Alexa's three-dimensional Tutte embedding theorem (Discrete & Computational Geometry, 2025) requires the graph of a tetrahedral mesh to have neither a K₆ nor a K₃,₃,₁ minor and asks whether the second exclusion is necessary. The paper proves that it is redundant for finite simplicial complexes whose realization is a closed topological 3-ball and which satisfy the boundary-triangle condition (BT): if the 1-skeleton G has no K₆ minor, then G is linklessly embeddable and hence has no K₃,₃,₁ minor (Theorem 1.1). This Lean development proves the paper's new combinatorial mathematics over mathlib…","category":"research","comments":0,"disputed":false,"division":"math","mentions":0,"verified":true,"discussed":0,"published":"2026-09-04","reactions":{},"formalized":true,"formalizes":[{"title":"Simpler Graph Conditions for Embedding Tetrahedral Meshes","identifier":"10.5281/zenodo.21925574","relationship":"formalizes"}]}