EFFICIENT IMBEDDINGS OF FINITE PROJECTIVE-PLANES

We use index-one voltage graph imbeddings of Cayley graphs (Cayley maps) for the groups Z(n2+n+1), where n is a prime power, to depict the finite projective planes PG(2, n). Subject to the condition that Z(n2+n+1) act regularly not only on the images of the points and of the lines of PG(2, n), but also on each orbit of the region set for the imbedding, the imbeddings constructed have maximum efficiency, as they attain the upper bound for the euler characteristic of an ambient space. The imbeddings are into orientable surfaces for n=1 or 2 (mod 3) and into orientable pseudosurfaces for n=0 (mod 3). An easy modification of the imbeddings for n=2 (mod 3) gives the genus of the (n+1, 6)-cages for these values. Efficient imbeddings are constructed also for the 'doubled' designs arising from the pairing of each PG(2, n) with its dual. The map automorphism groups are studied for all these imbeddings (lambda=1 and 2). Deletions from the imbeddings produce asymptotically efficient imbeddings for the affine planes AG(2, n).