Er --- (144 + 2 - 36 - 72)/2 = 19 . In fact, this almost certainly is the exact genus; and extends trivially to genus = n^2/2 + 1 for the n x n toric board, at any rate when n even. Fred Lunnon On 4/16/14, Fred Lunnon <fred.lunnon@gmail.com> wrote:
On 4/16/14, Adam P. Goucher <apgoucher@gmx.com> wrote:
Whoa! ... where does your 96 come from? WFL
There are 144 edges. The maximum number of faces is achieved when every face is a triangle, in which case there are (2/3).144 = 96 triangles.
Sincerely,
Adam P. Goucher
Ah, but there are no triangles: both graphs are bipartite, with girth 4 . So the number of faces equals 72 maximum; now via your reasoning, the genus of the larger graph equals at least (144 + 2 - 36 -72)/2 = 17 .
Also the genus is bounded above by that of the complete bipartite graph, which via Ringel's (1955) theorem equals Ceiling( (18-2)^2/4 ) = 64 .
Not such a nice surprise ...
Fred Lunnon