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All Journal Electronic Journal of Graph Theory and Applications (EJGTA)
Vladimir R. Rosenfeld
Department of Computer Science and Mathematics, Ariel University, Ariel 40700, Israel
Electrical & Electronics Engineering

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Traversing every edge in each direction once, but not at once: Cubic (polyhedral) graphs Vladimir R. Rosenfeld
Electronic Journal of Graph Theory and Applications (EJGTA) Vol 5, No 1 (2017): Electronic Journal of Graph Theory and Applications
Publisher : GTA Research Group, Univ. Newcastle, Indonesian Combinatorics Society and ITB

Show Abstract | Download Original | Original Source | Check in Google Scholar | DOI: 10.5614/ejgta.2017.5.1.13

Abstract

A {\em retracting-free bidirectional circuit} 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. Such a circuit revisits each vertex only in a number of steps. Studying the class $\mathit{\Omega}$ of all graphs admitting at least one retracting-free bidirectional circuit was proposed by Ore (1951) and is by now of practical use to nanotechnology. The latter needs in various molecular polyhedra that are constructed from a single chain molecule in the retracting-free way. Some earlier results for simple graphs, obtained by Thomassen and, then, by other authors, are specially refined by us for a cubic graph $Q$. Most of such refinements depend only on the number $n$ of vertices of $Q$.
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