The largest demigenus of a bipartite signed graph

A graph with signed edges is orientation embedded in a surface when it is topologically embedded so that one trip around a closed path preserves or reverses orientation according as the pathʹs sign product is positive or negative. We find the smallest surface within which it is possible to orientation-embed the complete bipartite signed graph ±Kr,s, which is obtained from the complete bipartite graph Kr,s through replacing each edge by two parallel edges, one positive and the other negative. We discuss some consequences and related problems.