Zigzags and central circuits for 3- or 4-valent plane graphs
Mathieu Dutour Sikirić, Michel Marie Deza · 2004
A central circuit in a 4-valent plane graph is a circuit of edges, such that no two consecutive edges belong to the same face. A zigzag in a 3- valent plane graph is a circuit of edges, such that any two, but no three, consecutive edges belong to the same face. Using the work of the Bielefeld school, we investigated the zigzag and central-circuit structure of 3- or 4-valent plane graphs, whose faces have gonality p or q. The case q=6 or 4 is of special interest, since it corresponds to faces having zero curvature. I will explain Thurston's formalism for handling those classes of graphs and how in the case of Goldberg- Coxeter construction, one can obtain the full description of the zigzag or central-circuit structure. Then, I will explain, the notion of tightness of graphs and the extremal problems related to it.