Shifts in Cayley graphs

An automorphism of an undirected simple graph is called a shift if it maps every vertex to an adjacent one. For all finite groups G , we determine the minimum nonzero valency of a Cayley graph on G that does not admit a shift. We also get a classification of groups with all involutions central and such that for every pair a , b of elements of the group, one of a b = b a , a b a − 1 = b − 1 , b a b − 1 = a − 1 or a 2 = b ± 2 holds.