Nonorientable genus of nearly complete bipartite graphs

Abstract

Let G(m, n, k), m, n≥3, k≤min(m, n), be the graph obtained from the complete bipartite graph Km,n by deleting an arbitrary set of k independent edges, and let {Mathematical expression} It is shown that the nonorientable genus {Mathematical expression}(G(m, n, k)) of the graph G(m, n, k) is equal to the upper integer part of f(m, n, k), except in trivial cases where f(m, n, k)≤0 and possibly in some extreme cases, the graphs G(k, k, k) and G(k + 1, k, k). These cases are also discussed, obtaining some positive and some negative results. In particular, it is shown that G(5, 4, 4) and G(5, 5, 5) have no nonorientable quadrilateral embedding. © 1988 Springer-Verlag New York Inc.