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

Italo J. Dejter

TL;DR

This work reframes the Hamiltonicity of odd graphs $O_k$ and middle-levels graphs $M_k$ through a castling tree on tight restricted-growth strings (TRGS) and their associated Dyck nests. By mapping TRGS to Dyck words via a casting procedure and blowing nests to fixed lengths, the author constructs anchored word representatives that encode cyclic and dihedral classes, enabling uniform 2-factors and modular arc factorizations that lift between $O_k$ and $M_k$. A clone/update mechanism on Dyck nests along the tree $T$ yields a scalable, combinatorial apparatus for generating Hamilton cycles, including explicit constructions for small $k$ and a general framework to obtain distinct Hamilton cycles from distinct spanning trees in hypergraphs. The approach offers an alternative perspective (via reversed complementation) on existing Hamiltonicity results, with detailed procedures for casting, blowing, anchoring, and assembling cycles that tie together Dyck-path combinatorics, Catalan structures, and Kneser-graph properties. Overall, the paper provides a cohesive, algorithmic pipeline from TRGS to Hamilton cycles in odd and middle-levels graphs, along with complexity bounds for the core castling operation and a suite of structural partitions helpful for SEO-friendly, semantically rich indexing.

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. Mütze 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. Mütze et al.

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

TL;DR

This work reframes the Hamiltonicity of odd graphs and middle-levels graphs through a castling tree on tight restricted-growth strings (TRGS) and their associated Dyck nests. By mapping TRGS to Dyck words via a casting procedure and blowing nests to fixed lengths, the author constructs anchored word representatives that encode cyclic and dihedral classes, enabling uniform 2-factors and modular arc factorizations that lift between and . A clone/update mechanism on Dyck nests along the tree yields a scalable, combinatorial apparatus for generating Hamilton cycles, including explicit constructions for small and a general framework to obtain distinct Hamilton cycles from distinct spanning trees in hypergraphs. The approach offers an alternative perspective (via reversed complementation) on existing Hamiltonicity results, with detailed procedures for casting, blowing, anchoring, and assembling cycles that tie together Dyck-path combinatorics, Catalan structures, and Kneser-graph properties. Overall, the paper provides a cohesive, algorithmic pipeline from TRGS to Hamilton cycles in odd and middle-levels graphs, along with complexity bounds for the core castling operation and a suite of structural partitions helpful for SEO-friendly, semantically rich indexing.

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 . From , a "blowing" operation recreates the whole family ol Dyck words. The vertices of can be elementarily updated all along . This simplifies an edge-supplementary arc-factorization view of Hamilton cycles of odd and middle-levels graphs found by T. Mütze 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. Mütze et al.
Paper Structure (14 sections, 17 theorems, 46 equations, 2 figures, 11 tables)

This paper contains 14 sections, 17 theorems, 46 equations, 2 figures, 11 tables.

Table of Contents

  1. Introduction
  2. Tight restricted-growth strings
  3. Tree of tight restricted-growth strings
  4. Dyck words, Dyck nests and blowing
  5. Castling procedure
  6. Dyck nests blown to a specific length
  7. Anchoring Dyck words to odd graphs
  8. Arc factorizations of odd graphs
  9. Permutations associated to Dyck words
  10. Uniform 2-factors of the odd graphs
  11. Partitions of odd-graph vertex sets
  12. Clones of Dyck nests
  13. Controlling odd and middle-levels graphs via T
  14. Hamilton cycles in odd and middle-levels graphs

Key Result

Lemma 21

Every Dyck nest $F=F^k=F(n)=F^k(n)$ of length $2k$ contains a substring $kk=k_1k_2$.

Figures (2)

  • Figure 1: Recovering $F(f)$ from $f$: $PLC(f)$ for triples (($f)$$[F(f)]$, $\underline{ord(f)})$, $k=1,2,3$
  • Figure 2: A section of $T$ accompanying Definitions \ref{['treee']}--\ref{['rho']} and Section \ref{['sec4']}

Theorems & Definitions (69)

  • Definition 1
  • Definition 2
  • Definition 3
  • Example 4
  • Definition 5
  • Definition 6
  • Example 7
  • Definition 8
  • Definition 9
  • Example 10
  • ...and 59 more