graph.famous {igraph}R Documentation

Creating named graphs

Description

There are some famous, named graphs, sometimes counterexamples to some conjecture or unique graphs with given features. These can be created with this function

Usage

graph.famous(name)

Arguments

name

Character constant giving the name of the graph. It is case insensitive.

Details

graph.famous knows the following graphs:

Bull

The bull graph, 5 vertices, 5 edges, resembles to the head of a bull if drawn properly.

Chvatal

This is the smallest triangle-free graph that is both 4-chromatic and 4-regular. According to the Grunbaum conjecture there exists an m-regular, m-chromatic graph with n vertices for every m>1 and n>2. The Chvatal graph is an example for m=4 and n=12. It has 24 edges.

Coxeter

A non-Hamiltonian cubic symmetric graph with 28 vertices and 42 edges.

Cubical

The Platonic graph of the cube. A convex regular polyhedron with 8 vertices and 12 edges.

Diamond

A graph with 4 vertices and 5 edges, resembles to a schematic diamond if drawn properly.

Dodecahedral, Dodecahedron

Another Platonic solid with 20 vertices and 30 edges.

Folkman

The semisymmetric graph with minimum number of vertices, 20 and 40 edges. A semisymmetric graph is regular, edge transitive and not vertex transitive.

Franklin

This is a graph whose embedding to the Klein bottle can be colored with six colors, it is a counterexample to the neccessity of the Heawood conjecture on a Klein bottle. It has 12 vertices and 18 edges.

Frucht

The Frucht Graph is the smallest cubical graph whose automorphism group consists only of the identity element. It has 12 vertices and 18 edges.

Grotzsch

The Grötzsch graph is a triangle-free graph with 11 vertices, 20 edges, and chromatic number 4. It is named after German mathematician Herbert Grötzsch, and its existence demonstrates that the assumption of planarity is necessary in Grötzsch's theorem that every triangle-free planar graph is 3-colorable.

Heawood

The Heawood graph is an undirected graph with 14 vertices and 21 edges. The graph is cubic, and all cycles in the graph have six or more edges. Every smaller cubic graph has shorter cycles, so this graph is the 6-cage, the smallest cubic graph of girth 6.

Herschel

The Herschel graph is the smallest nonhamiltonian polyhedral graph. It is the unique such graph on 11 nodes, and has 18 edges.

House

The house graph is a 5-vertex, 6-edge graph, the schematic draw of a house if drawn properly, basicly a triangle of the top of a square.

HouseX

The same as the house graph with an X in the square. 5 vertices and 8 edges.

Icosahedral, Icosahedron

A Platonic solid with 12 vertices and 30 edges.

Krackhardt\_Kite

A social network with 10 vertices and 18 edges. Krackhardt, D. Assessing the Political Landscape: Structure, Cognition, and Power in Organizations. Admin. Sci. Quart. 35, 342-369, 1990.

Levi

The graph is a 4-arc transitive cubic graph, it has 30 vertices and 45 edges.

McGee

The McGee graph is the unique 3-regular 7-cage graph, it has 24 vertices and 36 edges.

Meredith

The Meredith graph is a quartic graph on 70 nodes and 140 edges that is a counterexample to the conjecture that every 4-regular 4-connected graph is Hamiltonian.

Noperfectmatching

A connected graph with 16 vertices and 27 edges containing no perfect matching. A matching in a graph is a set of pairwise non-adjacent edges; that is, no two edges share a common vertex. A perfect matching is a matching which covers all vertices of the graph.

Nonline

A graph whose connected components are the 9 graphs whose presence as a vertex-induced subgraph in a graph makes a nonline graph. It has 50 vertices and 72 edges.

Octahedral, Octahedron

Platonic solid with 6 vertices and 12 edges.

Petersen

A 3-regular graph with 10 vertices and 15 edges. It is the smallest hypohamiltonian graph, ie. it is non-hamiltonian but removing any single vertex from it makes it Hamiltonian.

Robertson

The unique (4,5)-cage graph, ie. a 4-regular graph of girth 5. It has 19 vertices and 38 edges.

Smallestcyclicgroup

A smallest nontrivial graph whose automorphism group is cyclic. It has 9 vertices and 15 edges.

Tetrahedral, Tetrahedron

Platonic solid with 4 vertices and 6 edges.

Thomassen

The smallest hypotraceable graph, on 34 vertices and 52 edges. A hypotracable graph does not contain a Hamiltonian path but after removing any single vertex from it the remainder always contains a Hamiltonian path. A graph containing a Hamiltonian path is called tracable.

Tutte

Tait's Hamiltonian graph conjecture states that every 3-connected 3-regular planar graph is Hamiltonian. This graph is a counterexample. It has 46 vertices and 69 edges.

Uniquely3colorable

Returns a 12-vertex, triangle-free graph with chromatic number 3 that is uniquely 3-colorable.

Walther

An identity graph with 25 vertices and 31 edges. An identity graph has a single graph automorphism, the trivial one.

Zachary

Social network of friendships between 34 members of a karate club at a US university in the 1970s. See W. W. Zachary, An information flow model for conflict and fission in small groups, Journal of Anthropological Research 33, 452-473 (1977).

Value

A graph object.

Author(s)

Gabor Csardi csardi.gabor@gmail.com

See Also

graph can create arbitrary graphs, see also the other functions on the its manual page for creating special graphs.

Examples

solids <- list(graph.famous("Tetrahedron"),
               graph.famous("Cubical"),
               graph.famous("Octahedron"),
               graph.famous("Dodecahedron"),
               graph.famous("Icosahedron"))

[Package igraph version 0.6.5-1 Index]