Circular flows in mono-directed signed graphs

In this paper, the concept of circular $r$-flow in a mono-directed signed graph $\left(G,\sigma \right)$ is introduced. That is a pair $\left(D,f\right)$, where $D$ is an orientation on $G$ and $f:E\left(G\right)\to \left(-r,r\right)$ satisfies that $\mid f\left(e\right)\mid \in \left[1,r-1\right]$ for each positive edge $e$ and $\mid f\left(e\right)\mid \in \left[0,\frac{r}{2}-1\right]\cup \left[\frac{r}{2}+1,r\right)$ for each negative edge $e$, and the total in-flow equals the total out-flow at each vertex. This is the dual notion of circular colorings of signed graphs and is distinct from the concept of circular flows in bi-directed graphs associated with signed graphs studied in the literature. We first explore the connection between circular $\frac{2k}{k-1}$-flows and modulo $k$-orientations in signed graphs. Then we focus on the upper bounds for the circular flow indices of signed graphs in terms of the edge-connectivity, where the circular flow index of a signed graph is the minimum value $r$ such that it admits a circular $r$-flow. We prove that every 3-edge-connected signed graph admits a circular 6-flow and every 4-edge-connected signed graph admits a circular 4-flow. More generally, for $k\ge 2$, we show that every $\left(3k-1\right)$-edge-connected signed graph admits a circular $\frac{2k}{k-1}$-flow, every $3k$-edge-connected signed graph has a circular $r$-flow with $r<\frac{2k}{k-1}$, and every $\left(3k+1\right)$-edge-connected signed graph admits a circular $\frac{4k+2}{2k-1}$-flow. Moreover, the $\left(6k-2\right)$-edge-connectivity condition is shown to be sufficient for a signed Eulerian graph to admit a circular $\frac{4k}{2k-1}$-flow, and applying this result to planar graphs, we conclude that every signed bipartite planar graph of negative girth at least $6k-2$ admits a homomorphism to the negative even cycles $C_{-2k}$.