On a question of dirac on critical and vertex critical graphs

We give a construction which for any N provides a graph on n > N vertices which is vertex-critical with respect to being 4-chromatic, has at least cn² edges that are non-critical (i.e., the removal of any one does not change the chromaticity) and has at most Cn critical edges for some fixed positive constants c and C. Thus for any ε > 0 we get 4-vertex-critical graphs in which less than an ε-proportion of the edges are non-critical.