A signed bipartite (simple) graph $(G,\sigma )$ is said to be $C_{-4}$-critical if it admits no homomorphism to $C_{-4}$ (a negative 4-cycle) but each of its proper subgraphs does. To motivate the study of $C_{-4}$-critical signed graphs, we show that the notion of 4-coloring of graphs and signed graphs is captured, through simple graph operations, by the notion of homomorphism to $C_{-4}$. In particular, the 4-color theorem is equivalent to: Given a planar graph G, the signed bipartite graph obtained from G by replacing each edge with a negative path of length 2 maps to $C_{-4}$.

We prove that, except for one particular signed bipartite graph on 7 vertices and 9 edges, any $C_{-4}$-critical signed graph on n vertices must have at least $\lceil \frac{4n}{3}\rceil$ edges. Moreover, we show that for each value of $n\ge 9$ there exists a $C_{-4}$-critical signed graph on n vertices with either $\lceil \frac{4n}{3}\rceil$ or $\lceil \frac{4n}{3}\rceil +1$ many edges.

As an application, we conclude that all signed bipartite planar graphs of negative girth at least 8 map to $C_{-4}$. Furthermore, we show that there exists an example of a signed bipartite planar graph of girth 6 which does not map to $C_{-4}$, showing 8 is the best possible and disproving a conjecture of Naserasr, Rollova and Sopena.

带符号的二分（简单）图 $(G,\sigma )$ 据说是 $C_{-4}$-critical 如果它不承认同态 $C_{-4}$（负 4 循环）但它的每个适当的子图都是如此。为了激发研究$C_{-4}$- 临界有符号图，我们展示了通过简单的图操作，通过同态的概念来捕获图和有符号图的 4 着色概念 $C_{-4}$. 特别地，四色定理等价于： 给定一个平面图G ，通过用长度为 2 的负路径替换每条边从G获得的有符号二分图映射到$C_{-4}$.

我们证明，除了在 7 个顶点和 9 个边上的一个特定的有符号二部图外，任何 $C_{-4}$-n个顶点上的临界有符号图必须至少有$\lceil \frac{4n}{3}\rceil$边缘。此外，我们证明对于每个值$n\ge 9$ 存在一个 $C_{-4}$-n个顶点上的临界有符号图$\lceil \frac{4n}{3}\rceil$ 或者 $\lceil \frac{4n}{3}\rceil +1$ 许多边缘。

作为一个应用，我们得出结论，负周长的所有带符号二分平面图至少映射到 $C_{-4}$. 此外，我们表明存在一个周长为 6 的有符号二分平面图的例子，它没有映射到$C_{-4}$，显示 8 是最好的，并反驳了 Naserasr、Rollova 和 Sopena 的猜想。