The smallest surface that contains all signed graphs on K4,n

Finding the smallest number of crosscaps that suffice to orientation-embed every edge signature of the complete bipartite graph Km,n is an open problem. In this paper that number for the complete bipartite graph K4,n, n≥4, is determined by using diamond products of signed graphs. The number is 2⌊n−12⌋+1, which is attained by K4,n with exactly 1 negative edge, except that when n=4, the number is 4, which is attained by K4,4 with exactly 4 independent negative edges.