The chromatic number of a wheel graph (Wn) depends on whether the number of vertices in its outer cycle, n, is even or odd. Specifically, the chromatic number is 4 if n is even, and 3 if n is odd.
Understanding Wheel Graphs (Wn)
A wheel graph, denoted as Wn, is a type of graph constructed from a cycle graph (Cn) and an additional central vertex. This central vertex is connected to every single vertex of the Cn cycle. Therefore, a wheel graph Wn has a total of n + 1 vertices: n vertices forming the outer cycle and one central vertex. For example, W₃ would consist of a 3-vertex cycle and a central vertex connected to all three, which is isomorphic to a K₄ (complete graph with 4 vertices). For more comprehensive information on their structure and properties, you can explore the Wheel graph Wikipedia page.
Chromatic Number Defined
The chromatic number of a graph is a fundamental concept in graph theory. It represents the minimum number of colors required to color all the vertices of a graph such that no two adjacent vertices (vertices connected by an edge) share the same color. This process is known as a proper graph coloring.
Chromatic Number of Wn
The exact chromatic number for a wheel graph Wn is determined by the parity of n, the number of vertices in its defining cycle:
| Parity of n | Chromatic Number of Wn |
|---|---|
| Even | 4 |
| Odd | 3 |
This distinction highlights how the structure of the outer cycle influences the overall coloring requirements of the wheel graph, considering the central vertex's connections to all cycle vertices.
