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.

17 pages, 1 figure. Section 3.1 and Section 4 in the previous version is in Section 3 and Section 4 in the new version. Torsten Mütze pointed out that Theorem 3.1 was essentially proven in C.D.Savage, I. Shields and D. B. West, On the existence of Hamiltonian paths in the cover graph of M(n), Discrete Math, 262(1-3):241-252, 2003

References in corpus (1)

• (2+2)-free posets, ascent sequences and pattern avoiding permutations