Genus of K(n,t) for odd n

Determine the orientable genus γ(K(n,t)) of the graph K(n,t), defined as the complete graph K_n with a matching of size t deleted, when n is odd; in particular, ascertain γ(K(n,t)) for large values of t.

Background

The paper establishes genus formulas and constructions for K(n,t) when n is a multiple of 6, covering all graphs of minimum degree n−2 in that regime, and computes certain crossing numbers for complete graphs. However, for odd n the genus of K(n,t) is largely undetermined.

The author notes that, apart from a few small cases (e.g., γ(K(9,4)) = γ(K_9) = 3), the genus for odd n—especially for large t—remains largely unknown, highlighting this as a promising direction for future research.