Question

What is the chromatic index of each of the Platonic graphs?

Short answer

The chromatic index of the tetrahedral graph is 6, the cubic graph is 3, the octahedral graph is 4, the dodecahedral graph is 5, and the icosahedral graph is also 5.

Step-by-step solution

Identify the Platonic Graphs

The five Platonic graphs come from the five Platonic solids. These are the tetrahedral graph (denoted as T), cubic graph (C), octahedral graph (O), dodecahedral graph (D), and icosahedral graph (I). In these graphs, vertices represent the faces of the Platonic solid and edges represent the adjacency of these faces.

Chromatic Index of Tetrahedral Graph (T)

The tetrahedral graph has 4 vertices and 6 edges. Thus it can be seen that every edge touches every other edge. Therefore, the chromatic index of T is 6.

Chromatic Index of Cubic Graph (C)

A cube has 6 faces, so the cubic graph has 6 vertices and 12 edges. From the structure of a cubic graph where each vertex (edge on the Platonic solid) connects with three others, it can be deduced that the chromatic index of C is 3.

Chromatic Index of Octahedral Graph (O)

An octahedron has 8 faces, so the octahedral graph has 8 vertices and 12 edges. Every vertex in the graph is of degree 4, meaning that it takes at least 4 colors to color the edges. Thus, the chromatic index of O is 4.

Chromatic Index of Dodecahedral Graph (D)

A dodecahedron has 12 faces, so the dodecahedral graph has 12 vertices and 30 edges. Given that the dodecahedral graph is a regular graph of degree 5, it takes at least 5 colors to color the edges. Thus, the chromatic index of D is 5.

Chromatic Index of Icosahedral Graph (I)

An icosahedron has 20 faces, so the icosahedral graph has 20 vertices and 30 edges. This graph also requires at least 5 colors to properly color the edges. So, the chromatic index of I is 5.

Key concepts

Platonic Graphs

Platonic graphs are fascinating mathematical structures derived from Platonic solids. These solids are unique in 3D geometry because each face is a regular polygon, and they are highly symmetrical. There are five Platonic solids: the tetrahedron, cube, octahedron, dodecahedron, and icosahedron.

Each Platonic graph corresponds to one of these solids. In these graphs:

• Vertices represent the faces of the solid
• Edges represent the adjacency between faces

Understanding Platonic graphs involves analyzing both their geometric and topological properties, offering a beautiful intersection between geometry and graph theory.

Tetrahedral Graph

The tetrahedral graph serves as the graphical representation of a tetrahedron. A tetrahedron is a very simple Platonic solid, consisting of four triangular faces.

In its graphical form:

• There are 4 vertices.
• Each vertex connects to the other three, resulting in 6 edges.

Interestingly, every edge connects directly with every other edge, creating a complete graph known as the complete graph on four vertices, denoted as \( K_4 \). For the tetrahedral graph, its chromatic index is 6 because we need six distinct colors to uniquely color the edges without any two adjacent edges sharing the same color.

Cubic Graph

The cubic graph is associated with the cube, one of the most familiar Platonic solids, composed of six square faces. In graph theory terms:

• It has 6 vertices, correlating to the cube's faces.
• Each vertex connects to three others, leading to a total of 12 edges.

The cubic graph is structured such that each vertex is of degree 3—meaning any vertex connects to three other vertices. Because of this regular structure, its chromatic index is 3, corresponding to the number of colors needed to ensure no two adjacent edges share the same color.

Octahedral Graph

The octahedral graph corresponds to the octahedron. The octahedron is another polyhedron in the Platonic family with eight triangular faces. Its graph properties are as follows:

• The graph has 8 vertices.
• The vertex degree is 4, resulting in 12 edges overall.

In this graph, each vertex connects to four others, forming a symmetrical pattern that allows for easy coloring. To achieve optimal edge coloring, you need a chromatic index of 4, utilizing four unique colors to ensure that no two connected edges share the same color.

Dodecahedral Graph

The dodecahedral graph represents the dodecahedron, a Platonic solid made of twelve pentagonal faces. When represented as a graph:

• There are 12 vertices.
• The degree of each vertex is 5, forming 30 edges.

Due to its regular structure, where each vertex links five others, the graph is regular of degree 5. This connectivity requires a chromatic index of 5, meaning you need five distinct colors to properly color the edges, maintaining that no adjacent edges share the same color.

Icosahedral Graph

The icosahedral graph is derived from the icosahedron, a stunning Platonic solid with 20 triangular faces. The graph of this solid has the following characteristics:

• There are 20 vertices.
• Each vertex links to five others, contributing to the 30 edges present.

Within this graph, every vertex is of degree 5. The graph, being regular of degree 5, mandates a chromatic index of 5. This is the number of colors required to strategically color the edges so that no two touching edges share the same hue.