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

Abstract

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.