Problem 55 - Discrete Mathematics with Applications

Question

Give an example of a graph that is: Both Eulerian and Hamiltonian.

Step-by-Step Solution

Answer

The complete graph with 4 vertices (K4), where each vertex is connected to every other vertex, is an example of a graph that is both Eulerian and Hamiltonian. The Eulerian Circuit in K4 can be A-B-C-A-D-C-B-D-A, and the Hamiltonian Cycle can be A-B-C-D-A. K4 satisfies the necessary conditions for being Eulerian, as every vertex has an even degree, and meets the minimum degree criterion (at least 4/2) for Hamiltonian according to Dirac's theorem.

Step 1: Construct a graph satisfying Eulerian properties

We will start by constructing a graph with an even degree for every vertex. Let's create a complete graph with four vertices named A, B, C, and D, where each vertex is connected to every other vertex.

Step 2: Verify the graph's Hamiltonian properties

Now we need to verify if our graph is also Hamiltonian. We can use Dirac's theorem for this purpose. Our graph has 4 vertices and the minimum degree of each vertex is 3, which is at least 4/2. According to Dirac's theorem, our graph should be Hamiltonian.

Step 3: Show examples of the Eulerian Circuit and Hamiltonian Cycle

To further demonstrate the graph being both Eulerian and Hamiltonian, we will provide examples of Eulerian Circuit and Hamiltonian Cycle in our graph. Eulerian Circuit: A possible Eulerian Circuit in our graph would be A-B-C-A-D-C-B-D-A. Hamiltonian Cycle: A possible Hamiltonian Cycle in our graph would be A-B-C-D-A. In conclusion, the complete graph with 4 vertices (K4) is an example of a graph that is both Eulerian and Hamiltonian, as it satisfies the necessary conditions for being Eulerian and the minimum degree for Hamiltonian, and we were able to find an Eulerian Circuit and Hamiltonian Cycle in the graph.

Key Concepts

Graph Theory

Graph theory is the fascinating field of mathematics that deals with graphs, which are structures made up of vertices (or nodes) and edges connecting certain pairs of vertices. It is an essential component of discrete mathematics, providing the tools to study the connections and relationships between entities. Graphs are used in computer science, biology, transport networks, and many other fields.

• Vertices (Nodes): The fundamental units or points in a graph.
• Edges: The lines connecting pairs of vertices, signifying relationships.

The concepts of graph theory can vary in complexity, describing simple connections to more intricate paths or cycles. Understanding graph theory is crucial for analyzing problems in network design, pathway optimization, and efficient routing.

Eulerian Circuit

An Eulerian circuit, named after the mathematician Leonhard Euler, is a path in a graph that visits every edge exactly once and returns to the starting vertex. For a graph to contain an Eulerian circuit, it must satisfy a specific condition:

• Every vertex in the graph must have an even degree (the number of edges incident to a vertex).

This condition ensures that a journey through the graph can be completed without leaving any edge unvisited or re-tracing steps unnecessarily. A famous example of a problem rooted in Eulerian circuits is the "Seven Bridges of Königsberg," which laid foundational concepts for this area of study.

Hamiltonian Cycle

A Hamiltonian cycle is a closed loop on a graph where each vertex is visited exactly once before returning to the starting vertex. This cycle is distinct from an Eulerian circuit, which emphasizes edges rather than vertices. A Hamiltonian cycle must:

• Include each vertex exactly once.
• Start and end at the same vertex.

Determining whether a Hamiltonian cycle exists in a graph can be more complex because there is no simple condition like the Eulerian circuit's even-degree rule. However, some helpful tools, like Dirac's theorem, provide conditions under which such cycles can be guaranteed.

Dirac's Theorem

Dirac's theorem is a vital tool in determining whether a graph contains a Hamiltonian cycle. This theorem states that a simple graph with \( n \) vertices (where \( n \geq 3 \)) has a Hamiltonian cycle if every vertex has a degree of at least \( n/2 \).

• Simple Graph: A graph without loops or multiple edges between two vertices.
• Degree Requirement: Every vertex must connect to at least half of the total vertices in the graph.

Dirac's theorem provides a straightforward check for the existence of a Hamiltonian cycle, significantly simplifying the verification process for many graphs.

Complete Graph

A complete graph is a type of graph in which every pair of distinct vertices is connected by a unique edge. In a complete graph with \( n \) vertices, each vertex connects to \( n-1 \) other vertices. The notation for a complete graph is \( K_n \).

• Fully Connected: Each vertex has the highest possible degree, making the graph extremely interconnected.
• Simplicity: Complete graphs are among the simplest to analyze because their structure is highly predictable.

A complete graph with four vertices, denoted as \( K_4 \), is both Eulerian and Hamiltonian as each vertex has an even degree, and by Dirac's theorem, a Hamiltonian cycle exists. This property makes complete graphs perfect examples for studying Eulerian circuits and Hamiltonian cycles.