Deriving graphs with a retracting-free bidirectional double tracing

Abstract

A retracting-free bidirectional double tracing in a graph G is a closed walk which traverses every edge exactly once in each direction and such that no edge is succeeded by the same edge in the opposite direction. Studying the class Ω of all graphs admitting a retracting-free bidirectional double tracing was proposed by Ore (1951) and is, by now, of practical use to (bio)nanotechnology. In particular, this field needs various molecular polyhedra that are constructed from a single chain molecule in a retracting-free bidirectional double-tracing way. A cubic graph Q ∈ Ω has 3 h edges, where h is an odd number ≥3 . The graph of the triangular prism is the minimum cubic graph Q ∈ Ω , having 6 vertices and 9 edges. The graph of the square pyramid is the minimum polyhedral graph G in Ω , having 5 vertices and 8 edges. We analyze some possibilities for deriving new Ω -graphs from a given graph G ∈ Ω or G ∉ Ω using graph-theoretical operations. In particular, there was found that every noncycle Eulerian graph H admits a retracting-free bidirectional double tracing ( H ∈ Ω ) , which is a partial solution to the problem posed by Ore.