Blanuša snarks

{{Short description|Two 3-regular graphs with 18 vertices and 27 edges}}

{{infobox graph

| name = Blanuša snarks

| image = 220px

| image_caption = The first Blanuša snark

| namesake = Danilo Blanuša

| vertices = 18 (both)

| edges = 27 (both)

| chromatic_number = 3 (both)

| chromatic_index = 4 (both)

| diameter = 4 (both)

| radius = 4 (both)

| girth = 5 (both)

| automorphisms = 8, D₄ (1st)
4, Klein group (2nd)

| properties = Snark (both)
Hypohamiltonian (both)
Cubic (both)
Toroidal (only one){{cite journal | title=Blanuša double |author1=Orbanić, Alen |author2=Pisanski, Tomaž |author3=Randić, Milan |author4=Servatius, Brigitte|author4-link=Brigitte Servatius | journal=Math. Commun. | volume=9 | issue=1 | year=2004 | pages=91–103}}

|book thickness=3 (both)|queue number=2 (both)}}

In the mathematical field of graph theory, the Blanuša snarks are two 3-regular graphs with 18 vertices and 27 edges.{{MathWorld|title=Blanuša snarks|urlname=BlanusaSnarks}} They were discovered by Yugoslavian mathematician Danilo Blanuša in 1946 and are named after him.Blanuša, D., "Problem cetiriju boja." Glasnik Mat. Fiz. Astr. Ser. II. 1, 31-42, 1946. When discovered, only one snark was known—the Petersen graph.

As snarks, the Blanuša snarks are connected, bridgeless cubic graphs with chromatic index equal to 4. Both of them have chromatic number 3, diameter 4 and girth 5. They are non-hamiltonian but are hypohamiltonian.Eckhard Steen, "On Bicritical Snarks" Math. Slovaca, 1997. Both have book thickness 3 and queue number 2.Wolz, Jessica; Engineering Linear Layouts with SAT. Master Thesis, University of Tübingen, 2018

Algebraic properties

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 :

: $(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\$

: $(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).\$

Generalized Blanuša snarks

There exists a generalisation of the first and second Blanuša snark in two infinite families of snarks of order 8n+10 denoted $B_n^1$ and $B_n^2$ . The Blanuša snarks are the smallest members those two infinite families.Read, R. C. and Wilson, R. J. An Atlas of Graphs. Oxford, England: Oxford University Press, pp. 276 and 280, 1998.

In 2007, J. Mazák proved that the circular chromatic index of the type 1 generalized Blanuša snarks $B_n^1$ equals $3+{\frac {2} {n}}$ .J. Mazák, Circular chromatic index of snarks, Master's thesis, Comenius University in Bratislava, 2007.

In 2008, M. Ghebleh proved that the circular chromatic index of the type 2 generalized Blanuša snarks $B_n^2$ equals $3+{\frac {1} {\lfloor 1+3n/2\rfloor}}$ .M. Ghebleh, Circular Chromatic Index of Generalized Blanuša Snarks, The Electronic Journal of Combinatorics, vol 15, 2008.

Gallery

Image:First Blanusa snark 3COL.svg|The chromatic number of the first Blanuša snark is 3.

Image:First Blanusa snark 4edge color.svg|The chromatic index of the first Blanuša snark is 4.

Image:Second Blanusa snark 3COL.svg|The chromatic number of the second Blanuša snark is 3.

Image:Second Blanusa snark 4edge color.svg|The chromatic index of the second Blanuša snark is 4.

References

Category:Individual graphs

Category:Regular graphs