# Janoš Vidali: Cubic girth-regular graphs of small girth

Date: 19. 3. 2017

Source: Discrete mathematics seminar

Source: Discrete mathematics seminar

Torek, 21. 3. 2017, od 10h do 12h, Plemljev seminar, Jadranska 19

**Povzetek.** Let *Γ* be a *k*-regular graph of girth *g*. Suppose there exist
integers *s*_{1} ≤ *s*_{2} ≤ ... ≤ *s*_{k} such that for each vertex *u* of *Γ*, the *k*
arcs with initial vertex *u* lie on *s*_{1}, *s*_{2}, ... *s*_{k }girth cycles. Such a graph is said to be *girth-regular* with *signature* (*s*_{1}, *s*_{2}, ... *s*_{k}).
Girth-regular graphs include many classes of graphs, such as
vertex-transitive, semi-symmetric and distance-regular graphs, as well
as skeletons of many uniform maps and their truncations.

*a*,

*b*,

*c*). We then obtain classifications for girth-regular graphs with small and large values of

*c*, and for girth-regular graphs of girth at most 6. We also check for signatures appearing among vertex-transitive graphs of larger girths on at most 1280 vertices from the census by Potočnik, Spiga and Verret.