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

Hamiltonian cycles for the square of the augmentation graphs and Gray codes for restricted permutations and ascent sequences