This is a NEW EDITION using the condition that two edges incident with a vertex not in the dominating circuit cannot be attached to consecutive vertices in the dominating circuit.

I tried 100 million at random for each size $12, 16, \ldots, 48$ (it must be a multiple of 4).  They were all hamiltonian except for 4 graphs on 28 vertices.

Vertices $0,1,\ldots,20$ in natural order are the dominating cycle. The neighbours of the other vertices are as follows.

**Graph 1.**
 21 : 1 6 8;
 22 : 16 18 20;
 23 : 5 7 9;
 24 : 4 17 19;
 25 : 2 12 14;
 26 : 0 3 10;
 27 : 11 13 15;

**Graph 2.**
 21 : 1 3 5;
 22 : 13 15 17;
 23 : 9 14 16;
 24 : 7 12 20;
 25 : 0 2 4;
 26 : 6 10 19;
 27 : 8 11 18;

**Graph 3.**
 21 : 2 4 6;
 22 : 10 12 17;
 23 : 0 8 15;
 24 : 1 3 5;
 25 : 9 14 19;
 26 : 11 16 18;
 27 : 7 13 20;

**Graph 4.**
 21 : 3 17 19;
 22 : 4 18 20;
 23 : 1 5 16;
 24 : 2 6 14;
 25 : 8 10 12;
 26 : 9 11 13;
 27 : 0 7 15;

These have *not* been tested for 3-edge-colourability.