Chromatic numbers, Buchstaber numbers and chordality of Bier spheres

Limonchenko, Ivan (author), Vavpetič, Aleš (author)

We describe all the Bier spheres of dimension $d$ with chromatic number equal to $d+1$ and prove that all other $d$-dimensional Bier spheres have chromatic number equal to $d+2$, for any integer $d \ge 0$. Then we prove a general formula for complex and mod $p$ Buchstaber numbers of a Bier sphere ${\rm Bier}(K)$, for each prime $p \in {\mathbb N}$ in terms of the $f$-vector of the underlying simplicial complex $K$. Finally, we classify all chordal Bier spheres and obtain their canonical realizations as boundaries of stacked polytopes.

• Language: English
• Keywords: Bier sphere, Buchstaber number, chordal graph, chromatic number, stacked polytope
• Work type: Article
• Typology: 1.01 - Original Scientific Article
• Organization: FMF - Faculty of Mathematics and Physics
• Publication status: Published
• Publication version: Version of Record
• Publication date: 01.09.2026
• Year: 2026
• Number of pages: 16 str.
• Numbering: Vol. 349, iss. 9, art. 115189
• PID: 20.500.12556/RUL-182277
• UDC: 519.17:514
• ISSN on article: 0012-365X
• DOI: 10.1016/j.disc.2026.115189
• COBISS.SI-ID: 277148675
• Publication date in RUL: 06.05.2026

Record is a part of a journal

• Title: Discrete mathematics
• Shortened title: Discrete math.
• Publisher: Elsevier
• ISSN: 0012-365X
• COBISS.SI-ID: 1118479

Licences

• License: CC BY 4.0, Creative Commons Attribution 4.0 International
• Link: http://creativecommons.org/licenses/by/4.0/
• Description: This is the standard Creative Commons license that gives others maximum freedom to do what they want with the work as long as they credit the author.

Secondary language

• Language: Slovenian
• Keywords: Bierova sfera, Buchstaberjevo število, kromatično število, politopi

Projects

• Funder: ARIS - Slovenian Research and Innovation Agency
• Project number: P1-0292
• Name: Topologija in njena uporaba

• Funder: ARIS - Slovenian Research and Innovation Agency
• Project number: J1-4031
• Name: Računalniška knjižnica za zavozlane strukture in aplikacije