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An oriented walk double covering of a graph G is a set of oriented closed walks which, traversed successively, combined will have traced each edge of G once in each direction. A bidirectional double tracing of a graph G is an oriented walk double covering which consists of a single closed walk. A retracting in a closed walk is the immediate succession of an edge by its inverse. Every graph with minimum degree 2 has a retracting free oriented walk double covering and every connected graph has a bidirectional double tracing. The minimum number of closed walks in a retracting free oriented ∗This research was supported in part by the National Science Council of the Republic of China under grant NSC87-2115-110-004.

Oriented walk double covering and bidirectional double tracing