Concerning the recent notion of circular chromatic number of signed graphs, for each given integer $k$ we introduce two signed bipartite graphs, each on $2k^2-k+1$ vertices, having shortest negative cycle of length $2k$, and the circular chromatic number 4. Each of the construction can be viewed as a bipartite analogue of the generalized Mycielski graphs on odd cycles, $M_{\ell}(C_{2k+1})$. In the course of proving our result, we also obtain a simple proof of the fact that $M_{\ell}(C_{2k+1})$ and some similar quadrangulations of the projective plane have circular chromatic number 4. These proofs have the advantage that they illuminate, in an elementary manner, the strong relation between algebraic topology and graph coloring problems.

翻译：针对近期提出的带符号图圆色数概念，对于每个给定整数$k$，我们构造了两个带符号二分图，各有$2k^2-k+1$个顶点，且其最短负圈长度为$2k$，圆色数为4。每个构造可视为奇圈上广义Mycielski图$M_{\ell}(C_{2k+1})$的二分图类比。在证明过程中，我们还得到了$M_{\ell}(C_{2k+1})$及投影平面中某些类似四角剖分图具有圆色数4的简洁证明。这些证明的优势在于能以初等方式阐明代数拓扑与图着色问题之间的紧密联系。