  In this paper, we construct a listing for the vertices of the augmentation
graph of given size, and as a consequence, we obtain a Hamiltonian cycle for
the square of the augmentation graph of given size. As applications, we have a
Gray code for the $132$-$312$ avoiding permutations of given length such that
two successive permutations differ by at most $2$ adjacent transpositions. Also
we obtain Gray codes of strong distance $2$ for the $001$ avoiding ascent
sequences and the $010$ avoiding ascent sequences of given length.
