A cycle plus triangles graph is a 4-regular graph $G$ with a Hamiltonian circuit $C$ and such that the chords of $C$ induce a set of disjoint triangles (3-circuits). Our interest is in these graphs when they have an even number of triangles, so that the order of the graph is even. The chromatic index of these graphs is not always 4 (that is, they can't always be edge-colored with 4 colors). In the picture you see an example of a cycle plus triangles with an even number of triangles and chromatic index 5, so it is a class 2 graph.
The graph in the picture has an edge 2-cut (that is, two edges that if deleted make the graph fall apart). Our main question is the following:
Question 1 Is it possible to produce an example of a cycle plus triangles graph of even order which is class 2 and 3-connected, that is, so that it has no edge 2-cuts?
We have a side question (which might better belong to the cs sister site, but still)
Question 2 What is the complexity of deciding whether a cycle plus triangles graph of even order is class 1 or class 2?