A Cayley digraph Cay(G,S) of a finite group G with respect to a subset S
of G is said to be a CI-digraph if for every Cayley digraph Cay(G,T)
isomorphic to Cay(G,S), there exists an automorphism Ο of G such that
SΟ=T. A finite group G is said to have the m-DCI property for some
positive integer m if all m-valent Cayley digraphs of G are CI-digraphs,
and is said to be a DCI-group if G has the m-DCI property for all 1β€mβ€β£Gβ£. Let Q4nβ be a generalized quaternion group of order
4n with an integer nβ₯3, and let Q4nβ have the m-DCI
property for some 1β€mβ€2nβ1. It is shown in this paper that n is
odd, and n is not divisible by p2 for any prime pβ€mβ1. Furthermore,
if nβ₯3 is a power of a prime p, then Q4nβ has the m-DCI
property if and only if p is odd, and either n=p or 1β€mβ€p.Comment: 1