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Petersen's seminal work in 1891 asserts that the edge-set of a cubic graph can be covered by distinct perfect matchings if and only if it is bridgeless. Actually, it is known that for a very large fraction of bridgeless cubic graphs, every edge belongs to at least two distinct perfect matchings. In this paper, we study the class of non-double covered cubic graphs, i. e. \ graphs having an edge, called lonely edge, which belongs to exactly one perfect matching. First of all, we provide a reduction of the problem to the subclass U of 3-connected cubic graphs. Then, we furnish an inductive characterization of U and we study properties related to the count of lonely edges. In particular, denoting by Uₖ the subclass of graphs of U with exactly k lonely edges, we prove that Uₖ is empty for k>6, and we present a complete characterization for 3 k 6. The paper concludes with some insights on U₁ and U₂.
Goedgebeur et al. (Tue,) studied this question.