On Torsor Structures on Spanning Trees
Farbod Shokrieh, Cameron Wright · SIAM Journal on Discrete Mathematics · 2023
Abstract. We study two actions of the (degree 0) Picard group on the set of the spanning trees of a finite ribbon graph. It is known that these two actions, denoted [Formula: see text] and [Formula: see text], respectively, are independent of the base vertex [Formula: see text] if and only if the ribbon graph is planar. Baker and Wang [ Int. Math. Res. Not. IMRN, 16 (2018), pp. 5120–5147] conjectured that in a nonplanar ribbon graph without multiple edges, there always exists a vertex [Formula: see text] for which [Formula: see text]. We prove the conjecture and extend it to a class of ribbon graphs with multiple edges. We also give explicit examples exploring the relationship between the two torsor structures in the nonplanar case.