The following graphs are given in the graph6 format. The graphs were found using geng and SAGE.
All double-col-critical graphs with colouring number 6 and at most 10 vertices:
Order 1-6 (The complete graph on six vertices)
Order 7 (The graph obtained by adding one vertex v to the icosahedral graph I and joining v to each vertex of I)
Order 8 (7 graphs)
Order 9 (16 graphs)
Order 10 (91 graphs)