Blanuša snarks

The automorphism group of the first Blanuša snark is of order 8 and is isomorphic to the Dihedral group D₄, the group of symmetries of a square.

The automorphism group of the second Blanuša snark is an abelian group of order 4 isomorphic to the Klein four-group, the direct product of the Cyclic group Z/2Z with itself.

The characteristic polynomial of the first and the second Blanuša snark are respectively :

${\displaystyle (x-3)(x-1)^{3}(x+1)(x+2)(x^{4}+x^{3}-7x^{2}-5x+6)(x^{4}+x^{3}-5x^{2}-3x+4)^{2}\ }$

${\displaystyle (x-3)(x-1)^{3}(x^{3}+2x^{2}-3x-5)(x^{3}+2x^{2}-x-1)(x^{4}+x^{3}-7x^{2}-6x+7)(x^{4}+x^{3}-5x^{2}-4x+3).\ }$

There exists a generalisation of the first and second Blanuša snark in two infinite families of snarks of order 8n+10 denoted ${\displaystyle B_{n}^{1}}$ and ${\displaystyle B_{n}^{2}}$. The Blanuša snarks are the smallest members those two infinite families.^[7]

In 2007, J. Mazák proved that the circular chromatic index of the type 1 generalized Blanuša snarks ${\displaystyle B_{n}^{1}}$ equals ${\displaystyle 3+{\frac {2}{n}}}$.^[8]

In 2008, M. Ghebleh proved that the circular chromatic index of the type 2 generalized Blanuša snarks ${\displaystyle B_{n}^{2}}$ equals ${\displaystyle 3+{\frac {1}{\lfloor 1+3n/2\rfloor }}}$.^[9]

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  The chromatic number of the first Blanuša snark is 3.
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  The chromatic index of the first Blanuša snark is 4.
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  The chromatic number of the second Blanuša snark is 3.
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  The chromatic index of the second Blanuša snark is 4.