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Combinatorial Gray codes-an updated survey

Torsten Mütze

TL;DR

The survey comprehensively maps combinatorial Gray codes across diverse object classes, linking flip-graph Hamiltonicity to geometric and algebraic structures. It foregrounds general techniques (e.g., genlex order, greedy generation) and universal results (Sekanina/Fleischner, 0/1-polytopes) while updating the community on new sections (0/1-polytopes, ordered trees) and solved/open problems. Key contributions include a unified framework for Gray codes, broad algorithmic perspectives (constant-delay generation, ranking/unranking), and a curated open-problem collection (P-numbers) to stimulate further research. The work highlights the deep interplay between combinatorial reconfiguration, polytopal geometry, and classical graph-theoretic conjectures (Lovász, middle levels) with ramifications for efficient generation and enumeration in adjacent domains.

Abstract

A combinatorial Gray code for a class of objects is a listing that contains each object from the class exactly once such that any two consecutive objects in the list differ only by a `small change'. Such listings are known for many different combinatorial objects, including bitstrings, combinations, permutations, partitions, triangulations, but also for objects defined with respect to a fixed graph, such as spanning trees, perfect matchings or vertex colorings. This survey provides a comprehensive picture of the state-of-the-art of the research on combinatorial Gray codes. In particular, it gives an update on Savage's influential survey [C. D. Savage. A survey of combinatorial Gray codes. SIAM Rev., 39(4):605--629, 1997.], incorporating many more recent developments. We also emphasize the connections to closely related problems in graph theory, algebra, order theory, geometry and algorithms, which embeds this research area into a broader context. Lastly, we collect and propose a number of challenging research problems, thus stimulating new research endeavors.

Combinatorial Gray codes-an updated survey

TL;DR

The survey comprehensively maps combinatorial Gray codes across diverse object classes, linking flip-graph Hamiltonicity to geometric and algebraic structures. It foregrounds general techniques (e.g., genlex order, greedy generation) and universal results (Sekanina/Fleischner, 0/1-polytopes) while updating the community on new sections (0/1-polytopes, ordered trees) and solved/open problems. Key contributions include a unified framework for Gray codes, broad algorithmic perspectives (constant-delay generation, ranking/unranking), and a curated open-problem collection (P-numbers) to stimulate further research. The work highlights the deep interplay between combinatorial reconfiguration, polytopal geometry, and classical graph-theoretic conjectures (Lovász, middle levels) with ramifications for efficient generation and enumeration in adjacent domains.

Abstract

A combinatorial Gray code for a class of objects is a listing that contains each object from the class exactly once such that any two consecutive objects in the list differ only by a `small change'. Such listings are known for many different combinatorial objects, including bitstrings, combinations, permutations, partitions, triangulations, but also for objects defined with respect to a fixed graph, such as spanning trees, perfect matchings or vertex colorings. This survey provides a comprehensive picture of the state-of-the-art of the research on combinatorial Gray codes. In particular, it gives an update on Savage's influential survey [C. D. Savage. A survey of combinatorial Gray codes. SIAM Rev., 39(4):605--629, 1997.], incorporating many more recent developments. We also emphasize the connections to closely related problems in graph theory, algebra, order theory, geometry and algorithms, which embeds this research area into a broader context. Lastly, we collect and propose a number of challenging research problems, thus stimulating new research endeavors.
Paper Structure (72 sections, 1 equation, 17 figures)

This paper contains 72 sections, 1 equation, 17 figures.

Table of Contents

  1. Introduction
  2. Outline of this paper
  3. Preliminaries
  4. Flip graphs
  5. Necessary conditions
  6. Stronger Hamiltonicity notions
  7. General Gray code results
  8. Algorithmic questions
  9. Posets and polytopes
  10. 0/1-polytopes
  11. Lovász' conjecture and Cayley graphs
  12. Genlex ordering
  13. Binary strings
  14. The binary reflected Gray code
  15. Transition counts
  16. ...and 57 more sections

Figures (17)

  • Figure 1: Different flip graphs with a Hamilton path or cycle: (a) 4-cube $Q_4$; (b) Johnson graph $J(5,2)$; (c) Kneser graph $K(5,2)$ (=Petersen graph); (d) permutahedron; (e) sigma-tau graph (permutations under left shifts or swaps of the first two entries); (f) associahedron; (g) spanning trees of $K_4$ under edge exchanges.
  • Figure 2: Different binary Gray codes (1-bits black, 0-bits white): (a) BRGC $\Gamma_8$; (b) balanced; (c) monotone; (c') order-preserving; (d) long-run; (e) 8-antipodal; (e') 128-antipodal; (f) non-local; (g) trend-free; (h) Beckett; (i) $K_8$-code; (i') $Q_3$-code; (j) symmetric chains; (j') Greene/Kleitman symmetric chains; (k) single-track (length 240); (k') 2-track and 8-symmetric; (l) Savage-Shields-West (here $n=6$, unlike for the other codes where $n=8$).
  • Figure 3: Restricted binary Gray codes (1-bits black, 0-bits white): (a) coil-in-the-box (length 96); (b) 11-avoiding (Fibonacci words); (b') 11-avoiding circularly (Lucas words); (c) 101-avoiding; (c') 010-avoiding; (d1) necklaces (lex. largest); (d2) aperiodic necklaces (lex. largest); (d3) bracelets (lex. largest); (e1) necklaces (lex. smallest); (e1') unlabeled necklaces; (e2) Lyndon words (=lex. smallest aperiodic necklaces); (e2') unlabeled Lyndon words; (e3) weight $\leq 3$; (e4) 100-avoiding; (e5) $\leq 6$ inversions w.r.t. $0^*1^*$; (e6) $\leq 1$ transpositions w.r.t. $0^*1^*$; (e7) $\leq 00100101$; (e8) $\leq$ their reversal (lex. smallest neckties); (e9') $\{11,101\}$-avoiding; (e10') $\{100,101\}$-avoiding; (e11') 1-decreasing; (f) necklaces (repr. in BRGC order); (g) neckties (lex. smallest).
  • Figure 4: Venn diagram of combinatorial objects obtained from bitstrings under shifts, reversal and/or complementation.
  • Figure 5: Combination Gray codes (1-bits black, 0-bits white): (a) revolving door from BRGC; (b) homogeneous transpositions $00\cdots 01\leftrightarrow 10\cdots 00$; (c) transpositions $01\leftrightarrow 10$ or $001\leftrightarrow 100$; (d) adjacent transpositions (Eades-Hickey-Read); (d') adjacent transpositions (Ruskey); (e) star transpositions; (e') star transpositions + cyclic blocks; (f) alternating flaws; (g) prefix/substring shifts (cool-lex): (g1) all $(10,5)$-combinations, (g2) necklaces (lex. largest), (g3) aperiodic necklaces (lex. largest), (g4) 0111-avoiding, (g5) $\leq 10$ inversions w.r.t. $1^*0^*$, (g6) $\leq 2$ transpositions w.r.t. $1^*0^*$, (g7) $\geq 1001100010$, (g8) $\geq$ their reversal; (h) balanced $(17,2)$-combinations by transpositions with cyclic blocks.
  • ...and 12 more figures