Geodesic
Smallest snarks with oddness 4 and cyclic connectivity 4 have order 44
Ars Mathematica Contemporanea, Tome 16 (2019) no. 2, pp. 277-298.

Voir la notice de l'article provenant de la source Ars Mathematica Contemporanea website

The family of snarks – connected bridgeless cubic graphs that cannot be 3-edge-coloured – is well-known as a potential source of counterexamples to several important and long-standing conjectures in graph theory. These include the cycle double cover conjecture, Tutte’s 5-flow conjecture, Fulkerson’s conjecture, and several others. One way of approaching these conjectures is through the study of structural properties of snarks and construction of small examples with given properties. In this paper we deal with the problem of determining the smallest order of a nontrivial snark (that is, one which is cyclically 4-edge-connected and has girth at least 5) of oddness at least 4. Using a combination of structural analysis with extensive computations we prove that the smallest order of a snark with oddness at least 4 and cyclic connectivity 4 is 44. Formerly it was known that such a snark must have at least 38 vertices and one such snark on 44 vertices was constructed by Lukoťka, Máčajová, Mazák and Škoviera in 2015. The proof requires determining all cyclically 4-edge-connected snarks on 36 vertices, which extends the previously compiled list of all such snarks up to 34 vertices. As a by-product, we use this new list to test the validity of several conjectures where snarks can be smallest counterexamples.
DOI : 10.26493/1855-3974.1601.e75
Keywords: Cubic graph, cyclic connectivity, edge-colouring, snark, oddness, computation 
@article{10_26493_1855_3974_1601_e75,
     author = {Jan Goedgebeur and Edita M\'a\v{c}ajov\'a and Martin \v{S}koviera},
     title = {Smallest snarks with oddness 4 and cyclic connectivity 4 have order 44},
     journal = {Ars Mathematica Contemporanea},
     pages = {277--298},
     publisher = {mathdoc},
     volume = {16},
     number = {2},
     year = {2019},
     doi = {10.26493/1855-3974.1601.e75},
     language = {en},
     url = {http://geodesic.mathdoc.fr/articles/10.26493/1855-3974.1601.e75/}
}
TY  - JOUR
AU  - Jan Goedgebeur
AU  - Edita Máčajová
AU  - Martin Škoviera
TI  - Smallest snarks with oddness 4 and cyclic connectivity 4 have order 44
JO  - Ars Mathematica Contemporanea
PY  - 2019
SP  - 277
EP  - 298
VL  - 16
IS  - 2
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/articles/10.26493/1855-3974.1601.e75/
DO  - 10.26493/1855-3974.1601.e75
LA  - en
ID  - 10_26493_1855_3974_1601_e75
ER  - 
%0 Journal Article
%A Jan Goedgebeur
%A Edita Máčajová
%A Martin Škoviera
%T Smallest snarks with oddness 4 and cyclic connectivity 4 have order 44
%J Ars Mathematica Contemporanea
%D 2019
%P 277-298
%V 16
%N 2
%I mathdoc
%U http://geodesic.mathdoc.fr/articles/10.26493/1855-3974.1601.e75/
%R 10.26493/1855-3974.1601.e75
%G en
%F 10_26493_1855_3974_1601_e75
Jan Goedgebeur; Edita Máčajová; Martin Škoviera. Smallest snarks with oddness 4 and cyclic connectivity 4 have order 44. Ars Mathematica Contemporanea, Tome 16 (2019) no. 2, pp. 277-298. doi : 10.26493/1855-3974.1601.e75. http://geodesic.mathdoc.fr/articles/10.26493/1855-3974.1601.e75/

Cité par Sources :