  Topological properties of crystalline ice structures are studied by means of
self-avoiding walks on their H-bond networks. The number of self-avoiding
walks, C_n, for eight ice polymorphs has been obtained by direct enumeration up
to walk length n = 27. This has allowed us to determine the `connective
constant' or effective coordination number `mu' of these structures as the
limit of the ratio C_n/C_{n-1} for large n. This structure-dependent parameter
`mu' is related with other topological characteristics of ice polymorphs, such
as the mean and minimum ring size, or the topological density of network sites.
A correlation between the connective constant and the configurational entropy
of hydrogen-disordered ice structures is discussed.
