Wheel Graph

Definition

A wheel graph is a type of graph formed by connecting a single central vertex to all the vertices of a cycle. This creates a structure resembling a wheel, where the cycle forms the rim and the central vertex serves as the hub. The wheel graph is denoted as W_n, where n represents the total number of vertices in the graph, which includes both the central vertex and those on the cycle.

5 Must Know Facts For Your Next Test

1. The wheel graph W_n has n vertices and n edges, and it is formed by adding a central vertex to a cycle graph C_(n-1).
2. The chromatic number of a wheel graph W_n is 3 when n >= 4, which means at least three colors are needed to color its vertices without adjacent vertices sharing the same color.
3. For wheel graphs, if n = 3 (W_3), it is simply a triangle and can be colored with just 2 colors.
4. Wheel graphs are planar, meaning they can be drawn on a plane without edges crossing each other.
5. The degree of the central vertex in a wheel graph W_n is n - 1, while each vertex on the cycle has a degree of 2.

Review Questions

• How does the structure of a wheel graph influence its chromatic number?
  • The structure of a wheel graph significantly influences its chromatic number because of its unique combination of a central vertex connected to all vertices of a surrounding cycle. For any wheel graph W_n where n ≥ 4, at least three colors are necessary to ensure that adjacent vertices do not share the same color. The central vertex connects to all outer vertices, requiring an additional color to accommodate its connections, resulting in the need for three distinct colors for proper vertex coloring.
• Discuss why wheel graphs are considered planar graphs and how this property affects their chromatic number.
  • Wheel graphs are classified as planar because they can be drawn on a flat surface without any edges crossing each other. This planarity allows for straightforward visual representation, which aids in understanding their chromatic properties. Despite their planarity, the requirement for at least three colors arises from the connections between the central vertex and surrounding vertices, emphasizing that even planar graphs can have complex coloring requirements based on their structure.
• Evaluate how increasing the number of vertices in a wheel graph affects its chromatic number and overall properties.
  • As you increase the number of vertices in a wheel graph (i.e., moving from W_n to W_(n+1)), you maintain the chromatic number at 3 for n ≥ 4. However, this increase also enhances other properties like connectivity and complexity. The degree of the central vertex grows larger, providing more connections which can complicate certain algorithms used for coloring. Thus, while some fundamental characteristics remain constant, larger wheel graphs introduce new challenges in analyzing their properties and applying concepts like vertex coloring efficiently.

Related terms

Cycle Graph: A cycle graph is a graph that consists of a single cycle, meaning each vertex is connected in a closed loop with no branching paths.

Vertex Coloring: Vertex coloring is an assignment of labels (colors) to the vertices of a graph such that no two adjacent vertices share the same color.

Chromatic Number: The chromatic number is the smallest number of colors needed to color a graph's vertices so that no two adjacent vertices have the same color.