Marco Barbieri: Flexible 3-valent graphs of even girth

Flexible 3-valent graphs of even girth

Marco Barbieri (UL - FMF)

Abstract: Girth is a fascinating parameter in the study of symmetric graphs, since small girth often allows complete classification of the corresponding objects. Today, we focus on the existence of symmetric graphs of prescribed girth. In the 3-valent case, vertex-transitive graphs fall into three families according to the number of edge-orbits under their automorphism group. The existence problem has been settled for graphs with one and three edge-orbits, but remains open for the case of two. In this talk, I will sketch a proof establishing the existence of 3-valent vertex-transitive graphs with two edge-orbits and noncyclic stabilisers (the flexible case) and even girth, by reducing the problem to one in geometric group theory.