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 $cn^2$ 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 $\varepsilon>0$ we get $4$-vertex-critical graphs in which less than an $\varepsilon$-proportion of the edges are non-critical.