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

We show that for n = 4 and n ⩾ 6 , K n 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, K m ¯ + K n = K m + n − K m , 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 K m ¯ + G where m ⩾ | V ( G ) | − 1 . We provide a result that applies in some cases with smaller m when G is disconnected.