Leírás
An important class of matroids is graphic matroids, whose bases are the spanning trees. If we have a graph that is embedded into an orientable surface, then there is a natural notion similar to spanning
trees: A quasi-tree is a spanning subgraph that has one boundary component. Spanning trees are always quasi-trees, but if the embedding is into a higher genus surface, there might be quasi-trees that are not trees.
It turns out that quasi-trees also have nice combinatorial structure. This is captured by delta-matroids. In particular, embedded graphs have a delta-matroid, and the quasi-trees are the bases of the delta-matroid. A delta-matroid can be defined by a basis-exchange axiom that is very similar to the matroid basis exchange axiom. However, in a delta-matroid, the cardinality of the bases need not be the same. I will say some basic facts about delta matroids. Then, if there is time, I will talk about their sandpile groups. For a graph, the sandpile group is an Abelian group whose cardinality is the number of spanning trees. Recently, Merino, Moffatt and Noble defined an "embedded sandpile group" for embedded graphs, whose cardinality is the number of quasi-trees. We show that the embedded sandpile group is isomorphic to the usual sandpile group of the medial digraph of the graph.