Homomorphisms to small negative even cycles

A strengthening of Jaeger’s circular flow conjecture, restricted to planar graphs, asserts that every planar graph of odd girth at least 4k+1 admits a homomorphism to the odd cycle C2k+1, and the first case is verified and known as the famous Grötzsch theorem. In this paper, we prove analogous results for signed planar graphs: For k∈{2,3,4} every signed bipartite planar graph of negative girth at least 6k−4 admits a homomorphism to C−2k. Here the negative girth is the length of a shortest cycle with an odd number of negative edges. Note that the k=2 case was previously obtained in [J. Combin. Theory Ser. B, 153 (2022) 81–104] through a coloring method. Considering the duality between circular colorings and circular flows of planar graphs, our approach is based on the tools developed in the study of flows and group connectivity, and a potential method is applied in handling orientations with special boundaries for planar graphs. Furthermore, our results have several implications for the circular chromatic numbers of signed planar graphs with given girth conditions.