The nonorientable genus of the join of two cycles

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In this paper, we show that the nonorientable genus of C m + C n , the join of two cycles C m and C n , is equal to $\left\lceil {\tfrac{{(m - 2)(n - 2)}} {2}} \right\rceil $ if m = 3, n ≡ 1 (mod 2), or m ≥ 4, n ≥ 4, (m, n) ≠ (4, 4). We determine that the nonorientable genus of C 4 +C 4 is 3, and that the nonorientable genus of C 3 +C n is $\tfrac{n} {2} $ if n ≡ 0 (mod 2). Our results show that a minimum nonorientable genus embedding of the complete bipartite graph K m,n cannot be extended to an embedding of the join of two cycles without increasing the genus of the surface.