# CSM

## Seminar

### A bijection between subgraphs and orientations based on the combinatorics of the Tutte polynomial

Seminar Room 1, Newton Institute

#### Abstract

We present bijective correspondences between several structures on graphs. For any graph, we will describe a bijection between connected subgraphs and root-connected orientations, a bijection between spanning forests and score vectors and bijections between spanning trees, root-connected score vectors and recurrent sandpile configurations. These bijections are obtained as specializations of a general correspondence between spanning subgraphs and orientations of graphs. The definition and analysis of this correspondence rely on a recent characterisation of the Tutte polynomial and require to consider a \emph{combinatorial embedding} of the graph, that is, a choice of a cyclic order of the edges around each vertex.

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