Characterize graph properties that cause multiplicity-free multidegrees and nontrivial leading coefficients

Determine graph-theoretic conditions on a simple connected graph G that imply the multidegree polynomial C(G) of the binomial edge ideal J_G is multiplicity-free (all coefficients at most 1) and/or identify structural conditions that govern when the leading coefficient of C(G) equals 1 versus taking larger values.

Background

Using computational experiments (Macaulay2) on connected graphs up to nine vertices, the authors observed that a large proportion of multidegrees are multiplicity-free and that almost all have leading coefficient 1. They also exhibited a family (horned complete graphs) whose multidegree has leading coefficient equal to any prescribed positive integer n.

These observations motivate a structural understanding of which graph features enforce multiplicity-free behavior or determine the leading coefficient. The paper leaves this as an unresolved direction for future investigation.