Cuts in matchings of 3-edge-connected cubic graphs

We discuss relations between several known (some false, some open) conjectures on 3-edgeconnected, cubic graphs and how they all fit into the same framework related to cuts in matchings. We then provide a construction of 3-edge-connected digraphs satisfying the property that for every even subgraph E, the graph obtained by contracting the edges of E is not strongly connected. This disproves a recent conjecture of Hochstättler [A flow theory for the dichromatic number. European Journal of Combinatorics, 66, 160–167, 2017]. Furthermore, we show that an open conjecture of Neumann-Lara holds for all planar graphs on at most 26 vertices.