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

Authors:
M. N. Ellingham;D. Christopher Stephens
Affiliations:
Department of Mathematics, 1326 Stevenson Center, Vanderbilt University, Nashville, TN 37240, USA;Department of Mathematical Sciences, Middle Tennessee State University, Murfreesboro, TN 37132, USA
Venue:
Journal of Combinatorial Theory Series B
Year:
2007

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Abstract

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.