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 3h 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.
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PDFDOI: http://dx.doi.org/10.5614/ejgta.2022.10.2.1
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