Connection between Buchstaber invariants and generalized chromatic numbers

A. A. Aizenberg

Dalʹnevostočnyj matematičeskij žurnal, Tome 11 (2011) no. 2, pp. 113-139

Résumé

Let $K$ be a combinatorial simplicial complex. The work is devoted to $s(K)$ and $s_{\mathbb R}(K)$ – complex and real Buchstaber numbers which are combinatorial invariants of a simplicial complex $K$. The main tool in the study of these invariants is the notion of characteristic function, which can be regarded as an analogue of proper coloring in the graph theory. This analogy allows to prove estimations connecting Buchstaber numbers and chromatic number. It is shown in the work that real and complex Buchstaber numbers are not equal on the class of simplicial complexes. Also we provide an example of simplicial complexes $K$ and $L$ such that $s(K*L)\ne s(K)+s(L)$. The similarity between real Buchstaber number and chromatic number led to the notion of characteristic polynomial of simplicial complex. This polynomial takes values equal to the number of characteristic functions and possesses properties similar to those of a chromatic polynomial of a graph. The main results of the paper were reported on the section talk at the International conference “Toric Topology and Automorphic Functions” (September, 5–10th, 2011, Khabarovsk, Russia).