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：摘要：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部图，每个图包含$2k^2-k+1$个顶点，具有长度为$2k$的最短负环，且圆色数为4。每个构造可视为奇环上广义Mycielski图$M_{\ell}(C_{2k+1})$的二部图类比。在证明结论的过程中，我们同时得到了$M_{\ell}(C_{2k+1})$及若干类似的射影平面四边形剖分图圆色数为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染色问题之间的深层联系。