Polyhedral embeddings of snarks with arbitrary nonorientable genera (Q456335)

Summary: \textit{B. Mohar} and \textit{A. Vodopivec} [Comb. Probab. Comput. 15, No. 6, 877--893 (2006; Zbl 1108.05033)] proved that for every integer \(k\) (\(k \geq 1\) and \(k \neq 2\)), there exists a snark which polyhedrally embeds in \({\mathbb{N}}_k\) 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 \(k\) (\(k \geq 2\)), there exists an infinite family of snarks with nonorientable genus \(k\) which polyhedrally embed in \({\mathbb{N}}_k\). Furthermore, for every integer \(k\) (\(k> 0\)), there exists a snark with nonorientable genus \(k\) which polyhedrally embeds in \({\mathbb{N}}_k\).