In the

mathematical Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...

field of

graph theory In mathematics and computer science, graph theory is the study of ''graph (discrete mathematics), graphs'', which are mathematical structures used to model pairwise relations between objects. A graph in this context is made up of ''Vertex (graph ...

, the Blanuša snarks are two 3-

regular graph In graph theory, a regular graph is a Graph (discrete mathematics), graph where each Vertex (graph theory), vertex has the same number of neighbors; i.e. every vertex has the same Degree (graph theory), degree or valency. A regular directed graph ...

s with 18 vertices and 27 edges. They were discovered by

Yugoslavia , common_name = Yugoslavia , life_span = 1918–19921941–1945: World War II in Yugoslavia#Axis invasion and dismemberment of Yugoslavia, Axis occupation , p1 = Kingdom of SerbiaSerbia , flag_p ...

mathematician A mathematician is someone who uses an extensive knowledge of mathematics in their work, typically to solve mathematical problems. Mathematicians are concerned with numbers, data, quantity, mathematical structure, structure, space, Mathematica ...

Danilo Blanuša Danilo Blanuša (7 December 1903 – 8 August 1987) was a Croatian mathematician, physicist, engineer and a professor at the University of Zagreb. Biography Blanuša was born in Osijek, Austria-Hungary (today Croatia). He attended elementary schoo ...

in 1946 and are named after him. When discovered, only one snark was known—the

Petersen graph In the mathematical field of graph theory, the Petersen graph is an undirected graph with 10 vertices and 15 edges. It is a small graph that serves as a useful example and counterexample for many problems in graph theory. The Petersen graph i ...

. As

snark Snark may refer to: Fictional creatures * Snark (Lewis Carroll), a fictional animal species in Lewis Carroll's ''The Hunting of the Snark'' (1876) * Zn'rx, a race of fictional aliens in Marvel Comics publications, commonly referred to as "Snarks ...

s, the Blanuša snarks are connected, bridgeless

cubic graph In the mathematical field of graph theory, a cubic graph is a graph in which all vertices have degree three. In other words, a cubic graph is a 3-regular graph. Cubic graphs are also called trivalent graphs. A bicubic graph is a cubic bip ...

s with

chromatic index In graph theory, a proper edge coloring of a Graph (discrete mathematics), graph is an assignment of "colors" to the edges of the graph so that no two incident edges have the same color. For example, the figure to the right shows an edge colorin ...

equal to 4. Both of them have

chromatic number In graph theory, graph coloring is a methodic assignment of labels traditionally called "colors" to elements of a graph. The assignment is subject to certain constraints, such as that no two adjacent elements have the same color. Graph coloring i ...

3, diameter 4 and girth 5. They are non-hamiltonian but are hypohamiltonian. Both have

book thickness In graph theory, a book embedding is a generalization of planar embedding of a graph to embeddings in a ''book'', a collection of half-planes all having the same line as their boundary. Usually, the vertices of the graph are required to lie on ...

3 and queue number 2.

Algebraic properties

The

automorphism group In mathematics, the automorphism group of an object ''X'' is the group consisting of automorphisms of ''X'' under composition of morphisms. For example, if ''X'' is a finite-dimensional vector space, then the automorphism group of ''X'' is the g ...

of the first Blanuša snark is of order 8 and is

isomorphic In mathematics, an isomorphism is a structure-preserving mapping or morphism between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between the ...

to the

Dihedral group In mathematics, a dihedral group is the group (mathematics), group of symmetry, symmetries of a regular polygon, which includes rotational symmetry, rotations and reflection symmetry, reflections. Dihedral groups are among the simplest example ...

''D''₄, the group of symmetries of a square. The automorphism group of the second Blanuša snark is an

abelian group In mathematics, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on the order in which they are written. That is, the group operation is commu ...

of order 4 isomorphic to the

Klein four-group In mathematics, the Klein four-group is an abelian group with four elements, in which each element is Involution (mathematics), self-inverse (composing it with itself produces the identity) and in which composing any two of the three non-identi ...

, the

direct product In mathematics, a direct product of objects already known can often be defined by giving a new one. That induces a structure on the Cartesian product of the underlying sets from that of the contributing objects. The categorical product is an abs ...

of the

Cyclic group In abstract algebra, a cyclic group or monogenous group is a Group (mathematics), group, denoted C_n (also frequently \Z_n or Z_n, not to be confused with the commutative ring of P-adic number, -adic numbers), that is Generating set of a group, ge ...

Z/2Z with itself. The

characteristic polynomial In linear algebra, the characteristic polynomial of a square matrix is a polynomial which is invariant under matrix similarity and has the eigenvalues as roots. It has the determinant and the trace of the matrix among its coefficients. The ...

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 8''n''+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+

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

B_n^2

equals

3+

.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

of the first Blanuša snark is 3. Image:First Blanusa snark 4edge color.svg, The

of the first Blanuša snark is 4. Image:Second Blanusa snark 3COL.svg, The

of the second Blanuša snark is 3. Image:Second Blanusa snark 4edge color.svg, The

of the second Blanuša snark is 4.

References

{{DEFAULTSORT:Blanusa snarks Individual graphs Regular graphs