An infinite family of octahedral crossing numbers
Jonathan L. Gross · Journal of Graph Theory · 1978
Abstract This paper calculates some crossing numbers for certain octahedral graphs. Precisely, if the number p is a prime power congruent to 1 modulo 4, then the crossing number of the p‐dimensional octahedral graph in the orientable surface of genus (p–1)(p‐4)/4 is (p2–p)/2. The key step is the construction of a self‐dual imbedding of the complete graph on p vertices such that no face boundary contains a repeated vertex.