Polyhedral embeddings of snarks with arbitrary nonorientable genera

Abstract

Mohar and Vodopivec [Combinatorics, Probability and Computing (2006) 15, 877-893] proved that for every integer k (k≥1 and k≠2), there exists a snark which polyhedrally embeds in ℕk and presented the problem: Is there a snark that has a polyhedral embedding in the Klein bottle? In the paper, we give a positive solution of the problem and strengthen Mohar and Vodopivec's result. We prove that for every integer k (k≥2), there exists an infinite family of snarks with nonorientable genus k which polyhedrally embed in ℕk. Furthermore, for every integer k (k>0), there exists a snark with nonorientable genus k which polyhedrally embeds in ℕk.