From Surf Wiki (app.surf) — the open knowledge base
Blanuša snarks
Two 3-regular graphs with 18 vertices and 27 edges
Two 3-regular graphs with 18 vertices and 27 edges
| Field | Value | |
|---|---|---|
| name | Blanuša snarks | |
| image | [[Image:First Blanusa snark.svg | 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, *D*4 (1st) | |
| 4, Klein group (2nd) | ||
| properties | Snark (both) | |
| Hypohamiltonian (both) | ||
| Cubic (both) | ||
| Toroidal (only one) | ||
| book thickness | 3 (both) | queue number=2 (both) |
4, Klein group (2nd) Hypohamiltonian (both) Cubic (both) Toroidal (only one)
In the mathematical field of graph theory, the Blanuša snarks are two 3-regular graphs with 18 vertices and 27 edges. They were discovered by Yugoslavian mathematician Danilo Blanuša in 1946 and are named after him. 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. Both have book thickness 3 and queue number 2. Both graphs are 1-planar.{{citation | editor-first1 = Vida | editor-first2 = Fabrizio | editor-last1 = Dujmović | editor-last2 = Montecchiani | doi-access = free
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.
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}}.
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}}.
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
References
- (2004). "Blanuša double". [[Math. Commun.]].
- "Blanuša snarks".
- [[Danilo Blanuša. Blanuša, D.]], "Problem cetiriju boja." Glasnik Mat. Fiz. Astr. Ser. II. 1, 31-42, 1946.
- Eckhard Steen, "On Bicritical Snarks" Math. Slovaca, 1997.
- Wolz, Jessica; ''Engineering Linear Layouts with SAT.'' Master Thesis, University of Tübingen, 2018
- Read, R. C. and Wilson, R. J. An Atlas of Graphs. Oxford, England: Oxford University Press, pp. 276 and 280, 1998.
- J. Mazák, Circular chromatic index of snarks, Master's thesis, Comenius University in Bratislava, 2007.
- M. Ghebleh, Circular Chromatic Index of Generalized Blanuša Snarks, The Electronic Journal of Combinatorics, vol 15, 2008.
This article was imported from Wikipedia and is available under the Creative Commons Attribution-ShareAlike 4.0 License. Content has been adapted to SurfDoc format. Original contributors can be found on the article history page.
Ask Mako anything about Blanuša snarks — get instant answers, deeper analysis, and related topics.
Research with MakoFree with your Surf account
Create a free account to save articles, ask Mako questions, and organize your research.
Sign up freeThis content may have been generated or modified by AI. CloudSurf Software LLC is not responsible for the accuracy, completeness, or reliability of AI-generated content. Always verify important information from primary sources.
Report