Question

A wheel on \(n\) vertices consists of a cycle on \(n-1\) vertices together with one more vertex, normally drawn inside the cycle, that is connected to every vertex of the cycle. What is the chromatic number of a wheel on six vertices? What is the chromatic number of a wheel on an even number of vertices?

Short answer

The chromatic number of a wheel on six vertices is 3. For an even number of vertices, it's also 3.

Step-by-step solution

Understand the Structure of the Wheel Graph

The wheel on six vertices consists of a cycle graph, specifically a cycle with 5 vertices (since we exclude one vertex for the center). This cycle is often denoted as \( C_5 \). One additional vertex acts as the center, connecting to each of the 5 vertices in the cycle. Thus, the total number of vertices is 6, with the structure resembling a wheel.

Determine the Chromatic Number for Odd Cycle

A cycle with an odd number of vertices, such as \( C_5 \), requires 3 colors for proper coloring. This is because an odd cycle cannot be colored with just two colors without having two adjacent vertices share the same color.

Add the Center Vertex for Chromatic Number

Adding the center vertex, which is connected to all vertices in \( C_5 \), a third color must be used. This is because every adjacent vertex in the cycle already uses two different colors, requiring a third for the center.

Conclusion for Six Vertices Wheel Graph

For a wheel graph with six vertices, the chromatic number remains 3. This is due to the odd cycle \( C_5 \) within the graph that requires 3 colors, and the same or fewer colors suffice when adding one more vertex.

General Formulation for Even Vertices

A wheel graph with an even number of vertices has a cycle \( C_{n-1} \) where \( n-1 \) is even. An even cycle can be colored with 2 colors. When adding the central vertex, we need a third color, increasing the chromatic number to 3.

Final Conclusion

Thus, for a wheel graph on an even number of vertices, the chromatic number is always 3, following the same logic as adding a central vertex to an even cycle graph which requires two colors, while the center requires a third color.

Key concepts

Chromatic Number

In the world of graph theory, the chromatic number is crucial. It tells us the fewest colors needed to color a graph, ensuring no two adjacent vertices share the same color.
For a cycle graph with an odd number of vertices, like the cycle graph on 5 vertices denoted as \( C_5 \), the chromatic number is 3. This is because odd cycles are not bipartite, meaning they can't be colored with just two colors without creating a pair of adjacent vertices with the same color.
When discussing wheel graphs, which include an additional center vertex, this chromatic number calculation extends slightly. The center vertex connects to all vertices in the cycle, absorbing the existing two colors used, and thus requiring a third color.

• Odd cycle graph (e.g., \( C_5 \)): Uses 3 colors.
• Wheel graph: Typically demands one more color if the cycle graph part is already saturated with two colors.

Wheel Graph

A wheel graph can be visualized like a bicycle wheel composed of a central hub (center vertex) and spokes extending toward an outer ring (cycle graph). This graph structure is built from a cycle consisting of \( n-1 \) vertices and an additional center vertex that connects to all cycle graph vertices. The total vertex count becomes \( n \).

• Consists of a cycle and a central vertex.
• Central vertex connects to every point in the cycle, forming a web-like structure.

For a six-vertex wheel graph, like when \( n = 6 \), the core cycle has 5 vertices. Each vertex in the cycle connects to a central 6th vertex, hence its name "wheel graph." This makes it essential to understand the chromatic number for understanding coloring patterns: bring a new layer for graph coloring than cycle graphs alone.

Cycle Graph

A cycle graph consists simply of a closed loop of vertices, with each vertex connected exactly to two others, forming a single cycle. Cycle graphs, symbolically represented as \( C_n \), are foundational in graph theory. They can be either even or odd, determined by their number of vertices.

• Even-numbered cycle (\( C_{2k} \)): Requires only 2 colors.
• Odd-numbered cycle (\( C_{2k+1} \)): Needs 3 colors due to its non-bipartite nature.

In context, the cycle graph in a wheel graph sets the baseline for understanding how additional vertex and edge inclusions influence the chromatic number. For instance, knowing a \( C_5 \) cycle needs 3 colors sets the stage for understanding that adding a central vertex keeps the demand for 3 colors in a wheel graph with 6 vertices.