Vertex–Facet Assignment Obstructions Beyond Free Joins: Exact Defect, Transfer, and Constant-Additive Constructions
Year: 2026
Type: preprint
Abstract: Let P be a convex d-polytope with n vertices and m facets, and let N(P) be its vertex–facet nonincidence graph. For a face σ, let κP(σ) be the number of facets containing σ. Writing H(P) for the maximum of f0(σ)+κP(σ) over faces σ, we prove the exact matching formula ν(N(P))=n+m−H(P). The assignment criterion of Jahn and Winter is the vanishing of the resulting deficiency. A component decomposition separates deficiency within connected blocks from cancellation between blocks of opposite bipartite imbalance. We prove a transfer theorem for nondegenerate vertex-preserving one-point extensions in the strict visible/retained setting: connectedness of N(Q) is characterized by a rooted horizon graph, while surviving faces yield explicit matching bounds; in particular, strict stacking always produces a connected nonincidence graph. For every d≥7, this gives a rational balanced obstruction Rd of assignment deficiency one, with common class size 33 for d=7 and 3d+9 for d≥8. The seven-dimensional value is optimal within the factor-witnessed join–stack class. New balanced rational families have common size 2d+10 for even d≥18 and 2d+12 for odd d≥19, connected nonincidence graph, and matching number one below the common size. Hence the class optimum satisfies ηd=2d+O(1). Finally, for every d≥7 we construct infinitely many pairwise nonisomorphic rational obstructions and determine their matching numbers exactly. Exact-arithmetic programs audit all explicit incidence, matching, and connectivity certificates.
Authors: Wang, Qihang, Zhang, Dongming
Institution: Peking University
Language: English
Subfield: Discrete Mathematics and Combinatorics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7204094463
Cites:
Cited by:
Related to: