Toric ideal of matching polytopes and edge colorings
Kenta Mori, Ryo Motomura, Hidefumi Ohsugi, Akiyoshi Tsuchiya
TL;DR
The paper investigates the maximal degree of minimal generators ω(I_M_G) of toric ideals of matching polytopes and ties this to edge-coloring properties of multigraphs. It establishes an equivalence: ω(I_{S_G}) ≤ r iff any two k-vertex-colorings of replicated graphs G_{a} can be connected by a sequence of color-changes limited to r colors, providing a unifying framework that recovers known bipartite results and yields general bounds. A key contribution is the introduction of r-coloring ideals J_{G,r}, giving an explicit algebraic criterion for coloring equivalence and enabling systematic analysis of ω for stable and matching polytopes. The work further characterizes when ω(I_{M_G}) = 2 for line perfect graphs, proves ω ≤ 3 in that class, and explores ω in general graphs (including conjectures that ω ≤ 4) with computational and structural results that link to perfect graph theory and flow polytopes.
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.