Circular Flow Indices of Signed Graphs and Strongly Connected Orientations

This paper establishes a relation between orientations and circular flows in signed graphs. It is proved that the circular flow index of a signed graph is strictly smaller than if and only if it admits a strongly connected ‐extended‐Tutte orientation. As applications, we utilize a unified approach to show that every ‐edge‐connected signed planar graph has its circular flow index strictly less than for . By duality, this provides upper bounds for circular chromatic numbers of signed planar graphs with given girth conditions, improving several known results. In particular, our results imply that every signed planar graph of girth at least 8 has its circular chromatic number strictly less than 3.