Generalized Petersen graph

In graph theory, the generalized Petersen graphs are a family of cubic graphs formed by connecting the vertices of a regular polygon to the corresponding vertices of a star polygon. They include the Petersen graph and generalize one of the ways of constructing the Petersen graph. The generalized Petersen graph family was introduced in 1950 by H. S. M. Coxeter and was given its name in 1969 by Mark Watkins.
Definition and notation.
In Watkins' notation, "G"("n", "k") is a graph with vertex set
and edge set
where subscripts are to be read modulo "n" and "k" < "n"/2. Some authors use the notation "GPG"("n", "k"). Coxeter's notation for the same graph would be {"n"} + {"n"/"k"}, a combination of the Schläfli symbols for the regular "n"-gon and star polygon from which the graph is formed. The Petersen graph itself is "G"(5, 2) or {5} + {5/2}.
Any generalized Petersen graph can also be constructed from a voltage graph with two vertices, two self-loops, and one other edge.
Examples.
Among the generalized Petersen graphs are the "n"-prism "G"("n", 1), the Dürer graph "G"(6, 2), the Möbius-Kantor graph "G"(8, 3), the dodecahedron "G"(10, 2), the Desargues graph "G"(10, 3) and the Nauru graph "G"(12, 5).
Four generalized Petersen graphs – the 3-prism, the 5-prism, the Dürer graph, and "G"(7, 2) – are among the seven graphs that are cubic, 3-vertex-connected, and well-covered (meaning that all of their maximal independent sets have equal size).
Properties.
This family of graphs possesses a number of interesting properties. For example:
Isomorphisms.
"G"("n", "k") is isomorphic to "G"("n", "l") if and only if "kl" ≡ ±1 (mod "n").
Girth.
The girth of "G"("n", "k") is at least 3 and at most 8, in particular:
A table with exact girth values:
Chromatic number and chromatic index.
Being regular, according to Brooks' theorem their chromatic number can not be larger than their degree. Generalized Petersen graphs are cubic, so their chromatic number can be either 2 or 3. More exactly:
Where formula_5 denotes the logical AND, while formula_6 the logical OR. For example, the chromatic number of formula_7 is 3.
Petersen graph, being a snark, has a chromatic index of 4. All other generalized Petersen graph has chromatic index 3.
The generalized Petersen graph "G"(9, 2) is one of the few graphs known to have only one 3-edge-coloring.
The Petersen graph itself is the only generalized Petersen graph that is not 3-edge-colorable.