This paper classifies all snarks up to order $36$ and explains the reasons for their uncolourability. The crucial part of our approach is a computer-assisted structural analysis of cyclically $5$-connected critical snarks, which is justified by the fact that every other snark can be constructed from them by a series of simple operations while preserving uncolourability. Our results reveal that most of the analysed snarks are built from pieces of the Petersen graph and can be naturally distributed into a small number of classes having the same reason for uncolourability. This sheds new light on the structure of all small snarks. Based on our analysis, we generalise certain individual snarks to infinite families and identify a rich family of cyclically $5$-connected critical snarks.
@article{10_37236_10917,
author = {J\'an Maz\'ak and Jozef Rajn{\'\i}k and Martin \v{S}koviera},
title = {Morphology of small snarks},
journal = {The electronic journal of combinatorics},
year = {2022},
volume = {29},
number = {4},
doi = {10.37236/10917},
zbl = {1506.05076},
url = {http://geodesic.mathdoc.fr/articles/10.37236/10917/}
}
TY - JOUR
AU - Ján Mazák
AU - Jozef Rajník
AU - Martin Škoviera
TI - Morphology of small snarks
JO - The electronic journal of combinatorics
PY - 2022
VL - 29
IS - 4
UR - http://geodesic.mathdoc.fr/articles/10.37236/10917/
DO - 10.37236/10917
ID - 10_37236_10917
ER -
%0 Journal Article
%A Ján Mazák
%A Jozef Rajník
%A Martin Škoviera
%T Morphology of small snarks
%J The electronic journal of combinatorics
%D 2022
%V 29
%N 4
%U http://geodesic.mathdoc.fr/articles/10.37236/10917/
%R 10.37236/10917
%F 10_37236_10917
Ján Mazák; Jozef Rajník; Martin Škoviera. Morphology of small snarks. The electronic journal of combinatorics, Tome 29 (2022) no. 4. doi: 10.37236/10917
