Combinatorial Gray codes-an updated survey

TL;DR

This survey comprehensively reviews recent advances in combinatorial Gray codes, highlighting new developments, connections to other fields, and proposing future research challenges in the area.

Contribution

It updates Savage's 1997 survey with recent progress, broadening the scope and connecting Gray codes to various mathematical and algorithmic domains.

Findings

Recent developments in Gray code constructions for various combinatorial objects

Connections established between Gray codes and graph theory, algebra, and geometry

Identification of open problems and future research directions

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.