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 $\mathbb{N}_k$.
@article{10_37236_2539,
author = {Wenzhong Liu and Yichao Chen},
title = {Polyhedral embeddings of snarks with arbitrary nonorientable genera},
journal = {The electronic journal of combinatorics},
year = {2012},
volume = {19},
number = {3},
doi = {10.37236/2539},
zbl = {1252.05044},
url = {http://geodesic.mathdoc.fr/articles/10.37236/2539/}
}
TY - JOUR
AU - Wenzhong Liu
AU - Yichao Chen
TI - Polyhedral embeddings of snarks with arbitrary nonorientable genera
JO - The electronic journal of combinatorics
PY - 2012
VL - 19
IS - 3
UR - http://geodesic.mathdoc.fr/articles/10.37236/2539/
DO - 10.37236/2539
ID - 10_37236_2539
ER -
%0 Journal Article
%A Wenzhong Liu
%A Yichao Chen
%T Polyhedral embeddings of snarks with arbitrary nonorientable genera
%J The electronic journal of combinatorics
%D 2012
%V 19
%N 3
%U http://geodesic.mathdoc.fr/articles/10.37236/2539/
%R 10.37236/2539
%F 10_37236_2539
Wenzhong Liu; Yichao Chen. Polyhedral embeddings of snarks with arbitrary nonorientable genera. The electronic journal of combinatorics, Tome 19 (2012) no. 3. doi: 10.37236/2539
