Take the 2-minute tour ×  The cycle double covering conjecture
states that every bridgeless graph admits a collection of cycles
such that every edge in the graph is contained in exactly two
cycles from the collection (allowing repetitions). The
counterexample with the smallest number of edges is proved to be
cubic. My first question is that whether it is true that such a
minimal counterexample has the property that removing every edge
yields a bridge. In other words, is it true that if $G$ is a
graph with a cycle double covering, then $G\cup e$ also has a
cycle double covering, where $e$ is an added edge between two
vertices of $G$?  My second question is whether anything is
known regarding cycle coverings that cover every edge the same
number of times.  share|improve this question  1 Answer 1  up
vote 2 down vote accepted  No. In a minimal counterexample of
the cycle double cover conjecture, removing an edge creates no
bridges. The reason is simple. If such an edge existed, then
contracting it would yield a