Lower bounds on the length of universal traversal sequences

Universal traversal sequences for <italic>d</italic>-regular <italic>n</italic>-vertex graphs require length &OHgr;(<italic>d</italic><supscrpt>2</supscrpt><italic>n</italic><supscrpt>2</supscrpt> + <italic>dn</italic><supscrpt>2</supscrpt> log <italic>n/d</italic>), for 3 ≤ <italic>d</italic> ≤ <italic>n</italic>/3 - 2. This is nearly tight for <italic>d</italic> = &THgr;(<italic>n</italic>). We also introduce and study several variations on the problem, e.g. edge-universal traversal sequences, showing how improved lower bounds on these would improve the bounds given above.

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