The genus of Petersen powers

Abstract

For each k, 1≤k≤n, we construct a dot product of n copies of the Petersen graph whose orientable genus is precisely k. We show that these are all possible values for the genus of Pn. This result gives counterexamples of all possible genera to a conjecture of Tinsley and Watkins from 1982. We show that the Petersen graph is the only Petersen power which can be embedded into the projective plane. For each k, 2≤k≤n − 1, we construct a Petersen power Pn whose non-orientable genus is precisely k. © 2010 Wiley Periodicals, Inc. J Graph Theory 67:1-8, 2011