Leírás
EGERVÁRY SZEMINÁRIUM
Abstract: Let $\cal X$ be a given family on ground set E. We call a matroid on E an $\cal X$-matroid if each member of $\cal X$ is a matroid in it. The main problem considered in this talk is to decide whether a unique maximal $\cal X$-matroid exists under the weak order (where a matroid is larger than another one if its independent sets form a superset of the independent sets of the other one) and characterize this matroid. We consider this problem in the case where the ground set is the edge set of a graph G and $\cal X$ is formed by all subgraphs of G isomorphic to a given graph H (or, more generally, a graph H in a given set of graphs). We show several examples where the unique maximum exists, and also some other examples where there are more than one maximal matroids. We also present conjectures and open problems related to this problem. The talk is based on a recent paper of Jackson and Tanigawa and their joint papers with Clinch.