The Franklin graph is a 12-vertex graph (top picture) whose embedding in the Klein bottle (bottom image) splits it into areas with minimum coloring using six colors. References F. Apery. Real Projective Plane Models. Springer-Verlag, Berlin, 1986. H.S.M. Coxeter.

In mathematics, the Franklin graph is an undirected graph with 12 vertices and 24 edges, which can be embedded in the Klein bottle in such a way that each vertex lies on a separate face of the Klein bottle. The graph was introduced by Frank Adams in his article "A new kind of surface," Proceedings of the National Academy of Sciences 69: 2954-2957, 1972.

Franklin graphs are special cases of triangulations of **the projective plane**. That is, they are planar graphs that can be drawn in the plane without **edge crossings** so that every face is either a triangle or a quadrilateral. The Franklin graph has 6 triangles and 6 quadrilaterals faces. There are two ways to divide the edges of **a Franklin graph** between triangles and quadrilaterals so that each vertex lies on exactly three quadrilaterals and one triangle.

- How many vertices does a Klein bottle have?
- How many are simple non-isomorphic graphs on 4 vertices?
- Which figure has exactly six vertices?
- How many vertices does a cuboctahedron have?
- What is special about a Klein bottle?
- How many graphs are possible with 3 vertices?
- How many Hamilton circuits have six vertices?
- How many non-isomorphic trees are possible on 6 vertices?

11 graphs are known or conjectured to be simple, and they all have at least one vertex of degree 3. So there should be at most 11 different graphs with 4 vertices and no edges between any pair of vertices.

Here are the 11 graphs:

4 vertices: empty graph, complete bipartite graph K4, complete tripartite graph T4, four-hole graph H4

3 edges: triangle, tetrahedron, octahedron

6 Vertices in a Triangular Prism (corner points) - any three of which are equal.

The cuboctahedron, which has **8 triangle faces**, 6 square faces, 24 edges, and 12 vertices, is one of them. Another is the tiny rhombicuboctahedron, which has eight triangular faces, eight squares, 48 edges, and twenty-four vertices. It can be seen as a truncated cube with all but one face removed.

There are several other polyhedra that share some vertex numbers with the cuboctahedron. The first seven all have 8 square faces, and the last three all have 24: octahedron, tetradecahaedron, and dodecahedron. There are also six polyhedrons with 6 square faces and another six with 4 square faces. Finally, there are twelve polyhedrons with **2 square faces**.

It may help to know that most regular polyhedra are symmetric, meaning that they can be rotated or flipped over without changing **their shape** or surface. For example, the cuboctahedron is symmetric about each of **its axes** of symmetry. That means that if you were to rotate it any number of degrees, it would still look like a cuboctahedron.

As for the other regular polyhedra, the icosahedron is symmetric about all its axes of symmetry simultaneously, so it doesn't change when rotated.

The Klein bottle, like **the Mobius strip**, is a non-orientable two-dimensional manifold. The Klein bottle, unlike the Mobius strip, is a closed manifold, which means it is a compact manifold with **no border**. A sphere can be viewed as a special case of **a Klein bottle**, where the surface is orientable.

A Klein bottle may seem like an odd shape to us, but it has some interesting properties that make it worth studying:

It is one of **only five manifolds** that are not trivial to construct (the others being the three-sphere, projective space, and torus).

There is no continuous deformation from **the Klein bottle** to a plane, so it cannot be obtained by removing a neighborhood of a point or a line from the plane. This shows that the Klein bottle is a nontrivial topological object. Also, it cannot be constructed by attaching a 2-handle to the plane, since this would give rise to a manifold with boundary, not without boundary. This shows that the Klein bottle cannot be obtained by adding anything to a plane.

The Klein bottle can be constructed as the union of two copies of itself attached along their common boundary, which is also its unique circle of self-intersection.

We may deduce from this webpage that there are four unlabeled graphs on three vertices (indeed, the empty graph, an edge, a cherry, and the triangle). My solution is 8 graphs: for an undirected graph with no more than one edge between **any two nodes**. A graph with N vertices can only have n + 2 edges. Thus, we can create any of the eight graphs by adding either an edge or a cherry to the empty graph.

Example12.3

Number of vertices | Number of unique Hamilton circuits |
---|---|

5 | 12 |

6 | 60 |

7 | 360 |

8 | 2520 |

The six non-isomorphic trees of rank 6 are depicted in Figure 2. As stated in, Figure 3 depicts **the index value** and color codes of **the six trees** on their corresponding vertices. Neighbors are two vertices connected by an edge, and the degree of a vertex v in a graph G, indicated by degG (v), is the number of v's neighbors in G. Since all tree graphs have maximum degree three, this implies that trees can have **at most three pairs** of neighbors that are not adjacent.

In order to count the number of trees that can be drawn on any given set of vertices, we first need to determine how many distinct trees there are. A tree is defined as a connected graph with no cycles, and since graphs cannot have more than one cycle, this means that trees can have at most one cycle. A cycle is a sequence of edges connecting the same set of vertices, where each vertex appears exactly once in the sequence. Since trees can have zero or more than one cycle, they can have either zero or one of every length from 1 to n - 1 where n is the number of vertices in the tree. This means that there are n - 3 types of trees when n is greater than or equal to 4 and n - 2 types of trees when n is equal to 3.

A tree is called a forest if it has more than one tree.