Chromatic numbers, Buchstaber numbers and chordality of Bier spheres

Abstract

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≥0. Then we prove a general formula for complex and mod p Buchstaber numbers of a Bier sphere Bier(K), for each prime p∈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.