Circular flows in mono‐directed signed graphs

Abstract In this paper, the concept of circular ‐flow in a mono‐directed signed graph is introduced. That is a pair , where is an orientation on and satisfies that for each positive edge and for each negative edge , 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 ‐flows and modulo ‐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 such that it admits a circular ‐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 , we show that every ‐edge‐connected signed graph admits a circular ‐flow, every ‐edge‐connected signed graph has a circular ‐flow with , and every ‐edge‐connected signed graph admits a circular ‐flow. Moreover, the ‐edge‐connectivity condition is shown to be sufficient for a signed Eulerian graph to admit a circular ‐flow, and applying this result to planar graphs, we conclude that every signed bipartite planar graph of negative girth at least admits a homomorphism to the negative even cycles .