On the combinatorics of Murai spheres and its applications

We classify the combinatorial types of Murai spheres in dimensions $1$ and $2$, thereby showing that the corresponding convex simple polytopes have Delzant realizations. Then we describe all chordal Murai spheres ${\mathrm{B}\mathrm{i}\mathrm{e}\mathrm{r}}_c(M)$ with $c\in {\mathbb{N}}^m$ and $m\le 2$. Finally, we find all possible values for the Buchstaber and chromatic numbers of arbitrary Murai spheres.