  Given graphs $H_1, H_2$, a {red, blue}-coloring of the edges of a graph $G$
is a critical coloring if $G$ has neither a red $H_1$ nor a blue $ H_2$. A
non-complete graph $G$ is $(H_1, H_2)$-co-critical if $G$ admits a critical
coloring, but $G+e$ has no critical coloring for every edge $e$ in the
complement of $G$. Motivated by a conjecture of Hanson and Toft from 1987, we
study the minimum number of edges over all $(C_4, K_{1,k})$-co-critical graphs
on $n$ vertices. We show that for all $k \ge 2 $ and $ n \ge k +\lfloor \sqrt
{k-1} \rfloor +2$, if $G$ is a $(C_4,K_{1,k})$-co-critical graph on $n$
vertices, then \[e(G) \ge \frac{(k+2)n}2-3- \frac{(k-1)(k+ \lfloor \sqrt
{k-2}\rfloor)}2.\] Moreover, this linear bound is asymptotically best possible
for all $k\ge3$ and $n\ge3k+4$. It is worth noting that our constructions for
the case when $ k$ is even have at least three different critical colorings.
For $k=2$, we obtain the sharp bound for the minimum number of edges of $(C_4,
K_{1,2})$-co-critical graphs on $n\ge5 $ vertices by showing that all such
graphs have at least $2n-3$ edges. Our proofs rely on the structural properties
of $(C_4,K_{1,k})$-co-critical graphs and a result of Ollmann on the minimum
number of edges of $C_4$-saturated graphs.
