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

Abstract 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.