Universal sequences for graphs are studied. By letting U(d, n) denote the minimum length of a universal sequence for d-regular undirected graphs with n nodes, the latter paper has proved the upper bound U(d, n) = O(d2n3 log n) using a probabilistic argument. Here a lower bound of U(2, n) = Ω(n log n) is proved from which U(d, n) = Ω(n log n) for all d is deduced. Also, for complete graphs U(n - 1, n) = Ω(n log2 n/log n). An explicit construction of universal sequences for cycles (d = 2) of length nO(log n) is given.