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6-cycle double covers of cubic graphs

Rodrigo S. C. Leao·Valmir C. Barbosa
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Abstract

A cycle double cover (CDC) of an undirected graph is a collection of the graph's cycles such that every edge of the graph belongs to exactly two cycles. We describe a constructive method for generating all the cubic graphs that have a 6-CDC (a CDC in which every cycle has length 6). As an application of the method, we prove that all such graphs have a Hamiltonian cycle. A sense of direction is an edge labeling on graphs that follows a globally consistent scheme and is known to considerably reduce the complexity of several distributed problems. In [9], a particular instance of sense of direction, called a chordal sense of direction (CSD), is studied and the class of k-regular graphs that admit a CSD with exactly k labels (a minimal CSD) is analyzed. We now show that nearly all the cubic graphs in this class have a 6-CDC, the only exception being K4.

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Rodrigo S. C. Leao, Valmir C. Barbosa. 2009-04-17. 6-cycle double covers of cubic graphs. https://arxiv.org/abs/cs/0505088

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