Abstract
We answer a question on group connectivity suggested by Jaeger et al [J. Combin.
Theory, Ser. B 56 (1992), pp. 165-182]: we find that Z_2^2-connectivity does not imply Z_4-connectivity, neither vice versa.
We use a computer to find the graphs certifying this and to verify their properties using a nontrivial enumerative algorithm (and we also use an independent implementation of a straightforward algorithm to double-check our results). We provide a simple construction to provide an infinite family of examples.
While the graphs we found are small (the largest has 15 vertices and 21 edges), a computer-free approach remains elusive.