Question

Let \(G=(V, E)\) be a graph with \(|V| \geq 3\). Prove that if the degree of each vertex in \(G\) is at least \(|V| / 2\), then \(G\) is Hamiltonian.

Short answer

By Dirac's Theorem, \(G\) is Hamiltonian.

Step-by-step solution

Understanding the Problem

We are given a graph \(G=(V, E)\) with at least three vertices, i.e., \(|V| \geq 3\). Each vertex has a degree of at least \(|V| / 2\). Our goal is to prove that such a graph must be Hamiltonian, meaning it contains a Hamiltonian cycle, a cycle that visits each vertex exactly once and returns to the starting vertex.

Applying Dirac's Theorem

Dirac's Theorem states that a graph with \(|V| = n \geq 3\) is Hamiltonian if every vertex has degree at least \(n / 2\). This directly matches our problem's condition, as the degrees of all vertices in \(G\) are given to be at least \(|V| / 2\).

Conclusion from Dirac's Theorem

Since the condition of Dirac's Theorem is satisfied by the graph \(G\), we conclude that \(G\) is Hamiltonian. No further constructions or arguments are necessary because the theorem directly applies to and ensures the presence of a Hamiltonian cycle in \(G\).

Key concepts

Dirac's Theorem

In the world of graph theory, Dirac's Theorem is a crucial concept, especially when discussing Hamiltonian graphs. This theorem was proposed by the Danish mathematician Gabriel Dirac in 1952. It provides a clear, straightforward condition to determine the Hamiltonicity of a graph.

Dirac's Theorem states that a graph with at least three vertices (i.e., \(|V| = n \geq 3\)) is Hamiltonian if every vertex has a degree of at least \({n/2}\). The degree of a vertex is the number of edges connected to it. This criterion helps to predict the presence of a cycle that visits each vertex exactly once, also known as a Hamiltonian cycle.

Why is Dirac’s Theorem so significant?

• It simplifies the process of determining if a graph is Hamiltonian without needing to construct such a cycle explicitly.
• It applies to a broad range of graphs because the requirement of vertex degrees is relatively easy to check.

Graph theorists and mathematicians often rely on Dirac’s Theorem to make quick deductions about graphs, making it a cornerstone in graph theory studies.

Graph Theory

Graph theory is a vital area of mathematics that explores the relationships between objects. These objects are represented as vertices or nodes and are connected by edges. This field of study began with the famous Seven Bridges of Königsberg problem solved by Euler in 1736 and has since grown to have wide-ranging applications in computer science, biology, logistics, and social sciences.

In graph theory, various types of graphs are studied, such as

• Simple Graphs: Graphs that do not have multiple edges or loops.
• Directed Graphs (Digraphs): Graphs where edges have a direction, going from one vertex to another.
• Weighted Graphs: Graphs where edges have assigned weights, often representing distances or costs.

Graph theory also explores various concepts like connectivity, cycles, and paths. Understanding these basics helps in solving complex problems like finding the shortest path in a network or determining if a network is connected efficiently.

For students, having a firm grasp of graph theory fundamentals is essential to exploring more advanced topics such as network flows, coloring problems, and the intriguing Hamiltonian cycles.

Hamiltonian Cycle

A Hamiltonian cycle in a graph is a special type of cycle that visits every vertex exactly once and returns to the starting point. Identifying such a cycle can be important for numerous applications, such as solving routing problems or designing network circuits.

Here are some characteristics of Hamiltonian cycles:

• Not all graphs have Hamiltonian cycles, even if they appear complex or well-connected.
• Unlike Eulerian cycles, which visit every edge exactly once, Hamiltonian cycles focus on visiting vertices once.

Finding a Hamiltonian cycle can be challenging, and there is no simple algorithm for all graph types, largely making it a more complex problem than finding Eulerian paths or cycles.

Dirac’s Theorem significantly aids in identifying Hamiltonian cycles in certain graphs by providing a degree condition, ensuring that if each vertex meets or exceeds a specific degree, a Hamiltonian cycle is guaranteed to exist. This condition not only simplifies the process but also broadens our understanding of the structure necessary for such cycles to exist.

Understanding Hamiltonian cycles and Dirac’s Theorem together empowers students and graph theorists to tackle complex network-related problems with confidence.