Combinatorial Gray codes-an updated survey

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 elaborate on 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.

The Hamilton compression of highly symmetric graphs

“…Note that the hypercube is isomorphic to a Cayley graph of the abelian group Z n 2 . Hamilton cycles with various additional properties in the aforementioned families of graphs have been the subject of a long line of previous research under the name of combinatorial Gray codes [Sav97,Müt22]. We will see that some classical constructions of such cycles have a nontrivial small compression factor, and we construct cycles with much higher compression factor that we show to be optimal or near-optimal.…”

“…Hamilton cycles have been studied intensively from various different angles, such as graph theory (necessary/sufficient conditions, packing and covering etc. [Gou91,Gou03,Gou14,KO12]), optimization (shortest tours, approximation [ABCC06]), algorithms (complexity [GJ79], exhaustive generation [Sav97,Müt22]) and algebra (Cayley graphs [WG84, CG96, PR09, KM09]). In this work we introduce a new graph parameter that quantifies how symmetric a Hamilton cycle in a graph can be.…”

Structured Codes of Graphs

“…In [28], for example, Tonchev looked at the usual code distance problem restricted to codes whose codewords are characteristic vectors of edge sets of graphs. Gray codes on graphs are also considered, see [23], where the graphs representing the codewords should have some similarity properties if they are consecutive in a certain listing. Problems analogous to the present ones though restricted to special graph classes were also considered in [19] and [5].…”

Gray Cycles of Maximum Length Related to k-Character Substitutions

“…The so-called bubble languages [5,32] are also involved, as well as cross-bifix-free sets [4], Debruijn sequences [12], Dyck words [36], Fibonacci words [3], Lyndon words [39], Motzkin words [37], necklaces [31,39], set partitions [19], subsets of fixed size [9,15]: the list is far from exhaustive. For some surveys we suggest the reader report to [26,33,40,42]. From an algorithmic point of view, the ultimate feature is to develop methods for producing each new object with constant, or at least constant amortized time delay, that is loopless or constant amortized time algorithms are desired [10,24,35,38,41].…”

“…In addition, it is common in the literature to define a k-Gray code as the sequence where two consecutive items have distance at most k [26]. In the case of the Hamming distance, this notion corresponds to the so-called Σ k -Gray cycles, where the relation Σ k st. (w, w ′ ) ∈ Σ k iff.…”