Constructing generalized universal traversing sequences of polynomial size for graphs with small diameter

A generalized version of universal traversing sequences is constructed. The generalization preserves the features of the universal traversing sequences that make them attractive for applications to derandomizations and space-bounded computation. For every n, there is constructed a sequence that is used by a finite automaton with O(1) states in order to traverse all the n-vertex labeled undirected graphs. The automaton walks on the graph; when it is at a certain vertex, it uses the edge labels and the sequence in order to decide which edge to follow. When it is walking on an edge, the automaton can see the edge labeling. As a corollary, polynomial-size generalized universal traversing sequences constructible in DSpace(log n) are obtained for certain classes of graphs