Upper and lower bound theorems for graph-associahedra

TL;DR

This paper establishes optimal bounds for the gamma, g, h, and f vectors of graph-associahedra, expanding understanding of their combinatorial properties and providing inductive formulas for key invariants.

Contribution

It introduces new bounds for gamma-vectors of graph-associahedra and derives inductive formulas using a novel construction method combining shavings and differential equations.

Findings

Unimprovable bounds for gamma-vectors of graph-associahedra.

Construction method for higher-dimensional graph-associahedra via shavings.

Inductive formulas for gamma- and h-vectors of key polytope series.

Abstract

From the paper of the first author it follows that upper and lower bounds for $\gamma$-vector of a simple polytope imply the bounds for its $g$-,$h$- and $f$-vectors. In the paper of the second author it was obtained unimprovable upper and lower bounds for $\gamma$-vectors of flag nestohedra, particularly Gal's conjecture was proved for this case. In the present paper we obtain unimprovable upper and lower bounds for $\gamma$-vectors (consequently, for $g$-,$h$- and $f$-vectors) of graph-associahedra and some its important subclasses. We use the constructions that for an $(n-1)$-dimensional graph-associahedron $P_{\Gamma_n}$ give the $n$-dimensional graph-associahedron $P_{\Gamma_{n+1}}$ that is obtained from the cylinder $P_{\Gamma_n}\times I$ by sequential shaving some facets of its bases. We show that the well-known series of polytopes (associahedra, cyclohedra, permutohedra and stellohedra) can be derived by these constructions. As a corollary we obtain inductive formulas for $\gamma$- and $h$- vectors of the mentioned series. These formulas communicate the method of differential equations developed by the first author with the method of shavings developed by the second author.