Type: Seminar

Distribution: World

Expiry: 7 Jun 2013

CalTitle1: AGR Seminar: Four-valent graphs with a cross structure: Euler tours, chord diagrams, embeddings in surfaces

Auth: mathas@pmathas.pc (assumed)

**Host venue**

La Trobe University

**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**

Dr Yury Nikolayevsky

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If you would like to attend this seminar in our access grid room then please check to see if the grid is already booked at this time and send an email to accessgridroom@maths.usyd.edu.au to let the CSOs know that you would like to attend.