In 1978, Anderson and White asked whether there is a decomposition of K₁₂ into two graphs, one planar and one toroidal. Using theoretical arguments and a computer search of all maximal planar graphs of order 12, we show that no such decomposition exists. We further show that if G is planar of order 12 and H is toroidal, then H has at least two fewer edges than G. A computer search found all 123 unique pairs (G, H) that make this an equality. 9 pages
Bickle et al. (Thu,) studied this question.