The genera of edge amalgamations of complete bigraphs
Seth R. Alpert · Transactions of the American Mathematical Society · 1974
If G and H are graphs, then G ∨ H 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 ∨ K p , q ) g({K_{m,n}} \vee {K_{p,q}}) is either g ( K m , n ) + g ( K p , q ) g({K_{m,n}}) + g({K_{p,q}}) or else g ( K m , n ) + g ( K p , q ) − 1 g({K_{m,n}}) + g({K_{p,q}}) - 1 . The latter value is attained if and only if both K m , n {K_{m,n}} and K p , q {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.