Bhatia–Davis inequality: Difference between revisions

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{{infobox graph
| 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)
  | girth = 5 (both)
  | automorphisms = 8, [[Dihedral group|''D''<sub>4</sub>]] (1st)<br>4, [[Klein four-group|Klein group]] (2nd)
| properties = [[Snark (graph theory)|Snark]] (both)<br>[[Hypohamiltonian graph|Hypohamiltonian]] (both)<br>[[Cubic graph|Cubic]] (both)<br>[[Toroidal graph|Toroidal]] (only one)<ref>{{cite journal | title=Blanuša double | author = Orbanić, Alen; Pisanski, Tomaž; Randić, Milan; Servatius, Brigitte| journal=[[Math. Commun.]] | volume=9 | issue=1 | year=2004 | pages=91–103}}</ref>
}}
 
In the [[mathematics|mathematical]] field of [[graph theory]], the '''Blanuša snarks''' are two 3-[[regular graph]]s with 18 vertices and 27 edges.<ref>{{MathWorld|title=Blanuša snarks|urlname=BlanusaSnarks}}</ref> They were discovered by [[Croats|Croatian]] [[mathematician]] [[Danilo Blanuša]] in 1946 and are named after him.<ref>[[Danilo Blanuša|Blanuša, D.]], "Problem cetiriju boja." Glasnik Mat. Fiz. Astr. Ser. II. 1, 31-42, 1946.</ref> When discovered, only one snark was known—the [[Petersen graph]].
 
As [[Snark (graph theory)|snark]]s, the Blanuša snarks are connected, bridgeless [[cubic graph]]s with [[chromatic index]] equal to 4. Both of them have [[chromatic number]] 3, diameter 4 and girth 5. They are [[hamiltonian graph|non-hamiltonian]] but are [[Hypohamiltonian graph|hypohamiltonian]].<ref>Eckhard Steen, "On Bicritical Snarks" Math. Slovaca, 1997.</ref>
 
==Algebraic properties==
The [[automorphism group]] of the first Blanuša snark is of order 8 and is isomorphic to the [[Dihedral group]] ''D''<sub>4</sub>, 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 groups|direct product]] of the [[Cyclic group]] '''Z'''/2'''Z''' with itself.  
 
The [[characteristic polynomial]] of the first and the second Blanuša snark are respectively :
:<math>(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\ </math>
:<math>(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).\ </math>
 
==Generalized Blanuša snarks==
There exists a generalisation of the first and second Blanuša snark in two infinite families of snarks of order 8''n''+10 denoted <math>B_n^1</math> and <math>B_n^2</math>. The Blanuša snarks are the smallest members those two infinite families.<ref>Read, R. C. and Wilson, R. J. An Atlas of Graphs. Oxford, England: Oxford University Press, pp. 276 and 280, 1998.</ref>
 
In 2007, J. Mazak proved that the circular chromatic index of the type 1 generalized Blanuša snarks <math>B_n^1</math> equals <math>3+{\frac {2} {n}}</math>.<ref>J. Mazak, Circular chromatic index of snarks, Master's thesis, Comenius University in Bratislava, 2007.</ref>  
 
In 2008, M. Ghebleh proved that the circular chromatic index of the type 2 generalized Blanuša snarks <math>B_n^2</math> equals <math>3+{\frac {1} {\lfloor 1+3n/2\rfloor}}</math>.<ref>M. Ghebleh, Circular Chromatic Index of Generalized Blanuša Snarks, The Electronic Journal of Combinatorics, vol 15, 2008.</ref>
 
==Gallery==
<gallery>
Image:First Blanusa snark 3COL.svg|The [[chromatic number]] of the first Blanuša snark is&nbsp;3.
Image:First Blanusa snark 4edge color.svg|The [[chromatic index]] of the first Blanuša snark is&nbsp;4.
Image:Second Blanusa snark 3COL.svg|The [[chromatic number]] of the second Blanuša snark is&nbsp;3.
Image:Second Blanusa snark 4edge color.svg|The [[chromatic index]] of the second Blanuša snark is&nbsp;4.
</gallery>
 
== References ==
{{reflist}}
 
{{DEFAULTSORT:Blanusa snarks}}
[[Category:Individual graphs]]
[[Category:Regular graphs]]

Revision as of 14:19, 2 February 2014

Template:Infobox graph

In the mathematical field of graph theory, the Blanuša snarks are two 3-regular graphs with 18 vertices and 27 edges.[1] They were discovered by Croatian mathematician Danilo Blanuša in 1946 and are named after him.[2] 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.[3]

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 :

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 and . The Blanuša snarks are the smallest members those two infinite families.[4]

In 2007, J. Mazak proved that the circular chromatic index of the type 1 generalized Blanuša snarks equals .[5]

In 2008, M. Ghebleh proved that the circular chromatic index of the type 2 generalized Blanuša snarks equals .[6]

Gallery

References

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  2. ↑ Blanuša, D., "Problem cetiriju boja." Glasnik Mat. Fiz. Astr. Ser. II. 1, 31-42, 1946.
  3. ↑ Eckhard Steen, "On Bicritical Snarks" Math. Slovaca, 1997.
  4. ↑ Read, R. C. and Wilson, R. J. An Atlas of Graphs. Oxford, England: Oxford University Press, pp. 276 and 280, 1998.
  5. ↑ J. Mazak, Circular chromatic index of snarks, Master's thesis, Comenius University in Bratislava, 2007.
  6. ↑ M. Ghebleh, Circular Chromatic Index of Generalized Blanuša Snarks, The Electronic Journal of Combinatorics, vol 15, 2008.
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