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Generalized de Bruijn Cycles

Joshua N. Cooper·Ronald L. Graham
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Abstract

For a set of integers $I$, we define a $q$-ary $I$-cycle to be a assignment of the symbols 1 through $q$ to the integers modulo $q^n$ so that every word appears on some translate of $I$. This definition generalizes that of de Bruijn cycles, and opens up a multitude of questions. We address the existence of such cycles, discuss ``reduced'' cycles (ones in which the all-zeroes string need not appear), and provide general bounds on the shortest sequence which contains all words on some translate of $I$. We also prove a variant on recent results concerning decompositions of complete graphs into cycles and employ it to resolve the case of $|I|=2$ completely.

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Joshua N. Cooper, Ronald L. Graham. 2004-02-19. Generalized de Bruijn Cycles. https://arxiv.org/abs/math/0402324

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