Toric ideal of matching polytopes and edge colorings

TL;DR

This paper explores the algebraic structure of matching polytopes in graphs, linking toric ideal generators to edge coloring properties, and characterizes bipartite graphs with quadratic toric ideal generators.

Contribution

It establishes a connection between the degrees of toric ideal generators and edge coloring theory, providing new characterizations and conjectures for general graphs.

Findings

Toric ideal of bipartite graphs is generated by binomials of degree at most 3.

Characterization of bipartite graphs with quadratic toric ideal generators.

Discussion and conjecture on the maximal degree for general graphs.

Abstract

In the present paper, we investigate the maximal degree of minimal generators of the toric ideal of the matching polytope of a graph. It is known that the toric ideal associated to a bipartite graph is generated by binomials of degree at most $3$. We show that this fact is equivalent to a result in the theory of edge colorings of bipartite multigraphs. Moreover, a characterization of bipartite graphs whose toric ideals are generated by quadratic binomials is given. Finally, we discuss the maximal degree of minimal generators of the toric ideal associated to a general graph and give a conjecture.