Traversing every edge in each direction once, but not at once: Cubic (polyhedral) graphs

• Main Author: Vladimir R. Rosenfeld
• Format: Article
• Language: English
• Published: Indonesian Combinatorial Society (InaCombS); Graph Theory and Applications (GTA) Research Centre; University of Newcastle, Australia; Institut Teknologi Bandung (ITB), Indonesia 2017-04-01
• Series: Electronic Journal of Graph Theory and Applications
• Subjects: cubic graph, spanning tree, cotree, retracting-free bidirectional circuit
• Online Access: https://www.ejgta.org/index.php/ejgta/article/view/339

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 $Q$.