Nordhaus–Gaddum results for genus

We consider upper and lower bounds for γ ( G ) + γ ( G ¯ ) , the sum of the genus of a graph and its complement. For the lower bound, we show γ ( G ) + γ ( G ¯ ) ≥ ⌈ 1 12 ( n 2 − 13 n + 24 ) ⌉ . Furthermore, we construct an infinite family of graphs attaining this bound along with several other isolated examples. We provide a construction to show that γ ( G ) + γ ( G ¯ ) can be at least as large as 1 48 ( 5 n 2 − 52 n + 144 ) , and determine sharp upper bounds for a few small orders. Some asymptotic results are considered.