The genera of edge amalgamations of complete bigraphs

Abstract:

If G and H are graphs, then $G \vee H$ is defined to be a graph obtained by identifying some edge of G with some edge of H. It is shown that for all m, n, p, and q the genus $g({K_{m,n}} \vee {K_{p,q}})$ is either $g({K_{m,n}}) + g({K_{p,q}})$ or else $g({K_{m,n}}) + g({K_{p,q}}) - 1$. The latter value is attained if and only if both ${K_{m,n}}$ and ${K_{p,q}}$ are critical in the sense that the deletion of any edge results in a graph whose genus is one less than the genus of the original graph.