Counterexamples to a conjecture about bottlenecks in non-Tait-colourable cubic graphs

We follow the terminology and notion of [3]. By a well known theorem of Vizing it follows that the chromatic index z'(G) of a cubic graph G is 3 or 4. If z'(G) = 4 we say that G is non-Tait-colourable. Holroyd and Loupekine [1] defined a bottleneck in a non-Tait-colourable cubic graph G = (V,E) to be a minimal set ~ of odd cuts such that, for any subset S of E, either S or E\S contains at least one member of .~. They made the following conjecture: