# The $h^{*}$-polynomial of the cut polytope of $K_{2,m}$ in the lattice spanned by its vertices

## TL;DR

This paper explicitly computes the $h^{*}$-polynomial of the cut polytope for the complete bipartite graph $K_{2,m}$, advancing understanding of lattice polytope invariants in combinatorial optimization.

## Contribution

It provides the first explicit formula for the $h^{*}$-polynomial of the cut polytope of $K_{2,m}$ using Gröbner bases, filling a gap beyond trees.

## Key findings

- Explicit $h^{*}$-polynomial for $K_{2,m}$ cut polytope
- Uses Gröbner bases of toric ideals
- Advances classification of cut polytopes

## Abstract

The cut polytope of a graph is an important object in several fields, such as functional analysis, combinatorial optimization, and probability. For example, Sturmfels and Sullivant showed that the toric ideals of cut polytopes are useful in algebraic statistics. In the theory of lattice polytopes, the $h^{*}$-polynomial is one of the most important invariants. The necessary and sufficient condition in terms of graphs that the $h^{*}$-polynomial of a cut polytope is palindromic is known. However, except for trees, there are no classes of graphs for which the $h^{*}$-polynomial of their cut polytope is explicitly specified. In the present paper, we determine the $h^{*}$-polynomial of the cut polytope of complete bipartite graph $K_{2,m}$ using the theory of Gr\"{o}bner bases of toric ideals.