A New Look at the de Bruijn Graph

Fredricksen, H., A new look at the de Bruijn graph, Discrete Applied Mathematics 37138 (1992) 193-203. The Good-de Bruijn graph was originally defined to settle a question of existence of a certain shift register sequence, namely a binary cycle of length 2” containing each of the different binary n-tuples. The further properties of the graph have been studied by several authors. Because the graph is nonplanar and fairly complicated to draw for longer shift register lengths there have been several attempts to improve the description of the graph. We detail several of these attempts. Each description has its positive and negative features. Most assuredly each possible description of the graph illustrates some features and hides others. As the de Bruijn graph possesses dozens of interesting properties, each different presentation will have its own advantages. Finally, while considering an esoteric property that the graph possesses, the ultimate depiction of the graph, in the author’s view, has emerged. This version of the de Bruijn graph is the primary subject of the paper. “Would you tell me, please, which way I ought to go from here?” ‘*That depends a good deal on where you want to get to,” said the cat. “I don’t much care where-,” said Alice. “Then it doesn’t matter which way you go,” said the cat. Chapter 7, Alices Adventures in Wonderland