Polynomial universal traversing sequences for cycles are constructible

The paper constructs the first polynomial universal traversing sequences for cycles, solving an open problem of S. Cook and R. Aleliunas, R. Karp, R. Lipton, L. Lovasz, C. Rackoff (1979) [2] in the case of 2-regular graphs. The existence of universal traversing sequences of size <italic>&Ogr;</italic>(<italic>d</italic><supscrpt>2</supscrpt><italic>n</italic><supscrpt>3</supscrpt><italic>logn</italic>) for <italic>n</italic>-vertex <italic>d</italic>-regular graphs was established in [2] by a probabilistic argument, which was inherently non-constructive. For the cycles, the non-constructive upper bound was improved to <italic>&Ogr;</italic> (<italic>n</italic><supscrpt>3</supscrpt>) by Janowsky (1983) [13] and Cobham (1986) [8]. Previously, the best explicit constructions for cycles were due to Bridgland (1986) and A. Bar-Noy, A. Borodin, M. Karchmer, N. Linial, and M. Werman (1986), and have size <italic>&Ogr;</italic>(<italic>n<supscrpt>log n</supscrpt></italic>). Our universal traversing sequence has size <italic>&Ogr;</italic>(<italic>n</italic><supscrpt>4.76</supscrpt>), and can be constructed in log-space.