Universal sequences for complete graphs

An n-labeled complete digraph G is a complete digraph with n+1 vertices and n(n+1) edges labeled {1,2,...,n}_* such that there is a unique edge of each label emanating from each vertex. A sequence S in {1,2,...,n}_*: and a starting vertex of G define a unique walk in G, in the obvious way. Suppose S is a sequence such that for each such G and each starting point in it, the corresponding walk contains all the vertices of G. We show that the length of S is at least Ω(n²), improving a previously known Ω(nlog²_n/log logn) lower bound of Bar-Noy, Borodin, Karchmer, Linial and Werman.