On the excess of vertex-transitive graphs of given degree and girth

We consider a restriction of the well-known Cage Problem to the class of vertex-transitive graphs, and consider the problem of finding the smallest vertex-transitive k-regular graphs of girth g. Counting cycles to obtain necessary arithmetic conditions on the parameters (k,g), we extend previous results of Biggs, and prove that, for any given excess e and any given degree k≥4, the asymptotic density of the set of girths g for which there exists a vertex-transitive (k,g)-cage with excess not exceeding e is 0.