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 $k$ ($k \geq 1$ 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 $\mathbb{N}_k$.