Torsor Structures on Spanning Trees

2021/03/18 by Farbod Shokrieh, Cameron Wright, Shokrieh, Farbod +1
Computer Science · Mathematics · #05C10 #05C25 #Advanced Combinatorial Mathematics #Advanced Graph Theory Research #Combinatorics (math.CO) #FOS: Mathematics #Finite Group Theory Research

paper · pdf · doi:10.48550/arxiv.2103.10370

openalex publication_date 2021/03/18 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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 β_q and ρ_q respectively, are independent of the base vertex q if and only if the ribbon graph is planar. Baker and Wang conjectured that in a nonplanar ribbon graph without multiple edges there always exists a vertex q for which ρ_q≠β_q. 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.

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