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Confused about how to find de Bruijn sequence from Eulerian tour

I am not following how wikipedia constructs a de Bruijn sequence from an Eulerian tour here. When our Eulerian tour visits vertices $000,000, 001, 011, 111, 111, 110, 101, 011, 110, 100, 001, 010, 101,...
Princess Mia's user avatar
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Binary subsequence of size three, graph with euler cycle

Alan Tucker's Applied Combinatorics state, "A set of eight binary digits (0 or 1) are equally spaced about the edge of a disk. We want to choose the digits so that they form a circular sequence ...
Joceline Natali 's user avatar