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

TL;DR

This paper constructs Hamiltonian cycles in augmentation graphs and develops Gray codes for restricted permutations and ascent sequences, enabling efficient enumeration with minimal changes.

Contribution

It introduces a method to generate Hamiltonian cycles in augmentation graphs and creates Gray codes for specific restricted permutations and ascent sequences.

Findings

Hamiltonian cycle for the square of augmentation graphs constructed.

Gray code for 132-312 avoiding permutations with at most two transpositions.

Gray codes of strong distance 2 for 001 and 010 avoiding 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.