Castling tree of tight Dyck nests with applications to odd and middle-levels graphs

TL;DR

This paper introduces a novel 'castling' tree structure for tight Dyck words, linking combinatorial objects to Hamilton cycles in odd and middle-levels graphs, simplifying their analysis.

Contribution

It presents a new castling tree framework for tight Dyck words that clarifies the structure of Hamilton cycles in odd and middle-levels graphs.

Findings

Simplifies the understanding of Hamilton cycles in odd and middle-levels graphs.

Connects Dyck words with graph cycle structures through castling and blowing operations.

Provides a new combinatorial approach to analyze graph cycles.

Abstract

A subfamily of Dyck words called tight Dyck words is seen to correspond, via a "castling" procedure, to the vertex set of an ordered tree $T$. From $T$, a "blowing" operation recreates the whole family ol Dyck words. The vertices of $T$ can be elementarily updated all along $T$. This simplifies an edge-supplementary arc-factorization view of Hamilton cycles of odd and middle-levels graphs found by T. M\"utze et al. This take into account that the Dyck words represent: {\bf(a)} the cyclic and dihedral vertex classes of odd and middle-levels graphs, respectively, and {\bf(b)} the cycles of their 2-factors, as found by T. M\"utze et al.