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Let   $G$   be a  ribbon graph , that is, a connected finite graph   $G$   together with a cyclic ordering of the edges around each vertex. By adapting a construction due to Olivier Bernardi, we associate to any pair   $(v,e)$   consisting of a vertex   $v$   and an edge   $e$   adjacent to   $v$   a bijection   $\beta_{(v,e)}$   between spanning trees of   $G$   and elements of the set   $\operatorname{Pic}^g(G)$   of degree   $g$   divisor classes on   $G$  , where   $g$   is the  genus  of   $G$   in the sense of Baker–Norine. We give a new proof that the map   $\beta_{(v,e)}$   is bijective by explicitly constructing an inverse. Using the natural action of the Picard group   $\operatorname{Pic}^0(G)$   on   $\operatorname{Pic}^g(G)$  , we show that the Bernardi bijection   $\beta_{(v,e)}$   gives rise to a simply transitive action   $\beta_v$   of   $\operatorname{Pic}^0(G)$   on the set of spanning trees which does not depend on the choice of   $e$  . A  plane graph  has a natural ribbon structure (coming from the counterclockwise orientation of the plane), and in this case we show that   $\beta_v$   is independent of   $v$   as well. Thus for plane graphs, the set of spanning trees is naturally a torsor for the Picard group. Conversely, we show that if   $\beta_v$   is independent of   $v$   then   $G$   together with its ribbon structure is planar. We also show that the natural action of   $\operatorname{Pic}^0(G)$   on spanning trees of a plane graph is compatible with planar duality.

The Bernardi Process and Torsor Structures on Spanning Trees