Four-valent graphs with a cross structure: Euler tours, chord diagrams, embeddings in surfaces

Host Institution:

La Trobe University

Title of Seminar:

Four-valent graphs with a cross structure: Euler tours, chord diagrams, embeddings in surfacesbelian groups: their role in the pathological behaviour of braids and the downfall of democracy!

Speaker's Name:

Professor Denis Ilyutko

Speaker's Institution:

Moscow University, Russia

Time and Date:

Friday 7 June, 2.00 PM (AEST)

Seminar Abstract:

We consider finite connected four-valent graphs with a cross structure, i.e. graphs with a pairing of the four half-edges at each vertex. Graphs with a cross structure have Euler tours of different types depending on travelling through a vertex: we can pass from a half-edge to the opposite half-edge and we can pass from a half-edge to a non-opposite half-edge. In turn Euler tours are encoded by chord diagrams. There are criterion in terms of chord diagrams telling us when a graph with a cross structure can be embedded in the plane (Cairns–Elton and Read–Rosenstiehl) and surfaces with a genus g (Manturov). These criterion use different approaches depending on types of Euler tours in question. In the first part of the talk we consider a connection between these criterion. The second part of the talk is devoted to simple graphs and chord diagrams. It is known that there are graphs which are not circle graphs (not intersection graphs of chord diagrams) (Bouchet). But in spite of this fact many properties which circle graphs have remain true for simple graphs.

Seminar Convenor:

This email address is being protected from spambots. You need JavaScript enabled to view it. This email address is being protected from spambots. You need JavaScript enabled to view it.

AGR IT support:

This email address is being protected from spambots. You need JavaScript enabled to view it. This email address is being protected from spambots. You need JavaScript enabled to view it.