Signed bipartite circular cliques and a bipartite analogue of Grötzsch's theorem

The notion of the circular coloring signed graphs is a recent one that simultaneously extends both notions and $0$-free graphs. A $r$-coloring graph $(G, \sigma)$ to assign points circle circumference $r$, $r\geq 2$, vertices $G$ such connected by positive edge are at distance least $1$ negative most $\frac{r}{2}-1$. infimum all $r$ for which admits said be chromatic number denoted $\chi_c(G, \sigma)$. For any rational $r=\frac{p}{q}$, two cliques presented corresponding edge-sign preserving homomorphism switching homomorphism. It also shown restriction study numbers class bipartite simple already captures via basic operations, even though every bounded above $4$. In this work, we consider construct with respect homomorphisms. We then present reformulations $4$-Color Theorem Gr\"otzsch theorem. As analogue Gr\"otzsch's theorem, prove planar girth $6$ has $3$.