Mathematical Research Seminar - Archive

We will define and motivate a minor relation in the family of partial cubes. The main topic of the talk will be the forbidden minors characterizations of well established subfamilies. We will present new results on the family of partial cubes that have no forbidden minor isomorphic to a 3-dimensional hypercube minus a vertex. This family plays an important role in understanding bipartite graphs and in establishing a hierarchy among its subfamilies.

We consider a new type of regularity we call edge-girth-regularity. An edge-girth-regular \((v,k,g,\lambda)\)-graph \(G\) is a \(k\)-regular graph of order \(v\) and girth \(g\) in which every edge is contained in \(\lambda\) distinct \(g\)-cycles. This concept is a generalization of the well-known \((v,k,\lambda)\)-edge-regular graphs (that count the number of triangles) and appears in several related problems such as Moore graphs and Cage and Degree/Diameter Problems. All edge- and arc-transitive graphs are edge-girth-regular as well. We derive a number of basic properties of edge-girth-regular graphs, systematically consider cubic and tetravalent graphs from this class, and introduce several constructions that produce infinite families of edge-girth-regular graphs. We also exhibit several surprising connections to regular embeddings of graphs in orientable surfaces.

Joint work with Robert Jajcay and Štefko Miklavič.