Existence of a small cover over a 15-colorable simple 4-polytope

2023/02/09 by Djordje Baralić, Baralic, Djordje
Computer Science · Mathematics · #05C12 #52B11 #57S12 #Advanced Combinatorial Mathematics #Algebraic Topology (math.AT) #Combinatorics (math.CO) #Computational Geometry and Mesh Generation #FOS: Mathematics #Point processes and geometric inequalities

paper · pdf · doi:10.48550/arxiv.2302.04590

openalex publication_date 2023/02/09 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

The chromatic number for properly colouring the facets of a combinatorial simple n-polytope Pⁿ that is the orbit space of a quasitoric manifold satisfies the inequality n≤ Pⁿ≤ 2ⁿ-1. The inequality is sharp for n=2 but not for n=3 due to the Four Color theorem. In this note, we construct a simple 4-polytope admitting a characteristic map whose chromatic number equals 15 and deduce that the predicted upper bound is attained for n=4. Analogues results are verified for the case of oriented small covers in dimensions 4 and 5.

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