  Kostochka and Yancey proved that every $4$-critical graph $G$ has $e(G) \geq
\frac{5v(G) - 2}{3}$, and that equality holds if and only if $G$ is $4$-Ore. We
show that a question of Postle and Smith-Roberge implies that every
$4$-critical graph with no $(7,2)$-circular-colouring has $e(G) \geq
\frac{27v(G) -20}{15}$. We prove that every $4$-critical graph with no
$(7,2)$-colouring has $e(G) \geq \frac{17v(G)}{10}$ unless $G$ is isomorphic to
$K_{4}$ or the wheel on six vertices. We also show that if the Gallai Tree of a
$4$-critical graph with no $(7,2)$-colouring has every component isomorphic to
either an odd cycle, a claw, or a path. In the case that the Gallai Tree
contains an odd cycle component, then $G$ is isomorphic to an odd wheel. In
general, we show a $k$-critical graph with no $(2k-1,2)$-colouring that
contains a clique of size $k-1$ in it's Gallai Tree is isomorphic to $K_{k}$.
