Nonorientable genus embedding of nearly complete bipartite graphs

The nearly complete bipartite graph $G(m,n,k)$ is obtained by removing $k$ independent edges from the complete bipartite graph $K_{m,n}$. In this paper, we prove that for any nearly complete bipartite graph $G(m,n,k)$ with $m, n\geq 3$, and $(m,n,k)\notin\{(5,4,4)$, $(4,5,4)$, $(5,5,5)\}$, there exists a nonorientable genus embedding $\Pi$ satisfying $\tilde{\gamma}(\Pi)=\max\{\lceil \big((m-2)(n-2)-k\big)/2\rceil, 1\}$. This embedding can be constructed by starting from an embedding of some $G(p,q,h)$ with $h\leq 6$ and $p,q\leq 7$, and then iteratively adding multiple copies of $G(2,2,2)$, $G(2,0,0)$ and $G(0,2,0)$. As a consequence, the previously unresolved nonorientable genus $\tilde{\gamma}(G(n+1,n,n))$ for even $n$ and $\tilde{\gamma}(G(n,n,n))$ for arbitrary $n$ are now determined.