The nonorientable genus of joins of complete graphs with large edgeless graphs

We show that for n=4 and n⩾6, Kn has a nonorientable embedding in which all the facial walks are hamilton cycles. Moreover, when n is odd there is such an embedding that is 2-face-colorable. Using these results we consider the join of an edgeless graph with a complete graph, , and show that for n⩾3 and m⩾n−1 its nonorientable genus is ⌈(m−2)(n−2)/2⌉ except when (m,n)=(4,5). We then extend these results to find the nonorientable genus of all graphs where m⩾|V(G)|−1. We provide a result that applies in some cases with smaller m when G is disconnected.