Problem 43 - Discrete Mathematics with Applications

Question

Under what conditions will the complete graph \(K_{n}\) be Hamiltonian?

Step-by-Step Solution

Answer

A complete graph \(K_n\) is Hamiltonian under the conditions of Ore's Theorem and Dirac's Theorem. Since every vertex in a complete graph is connected to every other vertex with degree (n-1), which is always greater than or equal to n/2, both theorems are satisfied, and a Hamiltonian cycle exists within the graph.

Step 1: Understanding Complete Graphs

A complete graph is a simple graph in which every pair of distinct vertices is connected by a unique edge. In other words, every vertex is connected to every other vertex. The complete graph with "n" vertices is denoted by \(K_n\) and has \(\frac{n(n-1)}{2}\) edges.

Step 2: Observing Hamiltonian Cycle in Complete Graph

A Hamiltonian cycle is a cycle that passes through all vertices exactly once and returns to the starting vertex. A Hamiltonian cycle can visit every vertex in a complete graph \(K_n\) because all vertices are connected to every other vertex in the graph.

Step 3: Apply Conditions for Hamiltonian Graphs

According to Ore's Theorem or Dirac's Theorem, a graph G is Hamiltonian if: 1. Ore's Theorem: For every pair of non-adjacent vertices, the sum of their degrees is greater than or equal to n. Since all vertices are connected in a complete graph, this condition is satisfied. 2. Dirac's Theorem: If every vertex in G has a degree greater than or equal to n/2, then G is Hamiltonian. In a complete graph \(K_n\), every vertex is connected to every other vertex, so the degree of each vertex is (n-1). Since (n-1) is always greater than or equal to n/2, the complete graph \(K_n\) satisfies this condition.

Step 4: Conclusion

Since the conditions in Ore's Theorem and Dirac's Theorem are satisfied for a complete graph \(K_n\), it can be concluded that every complete graph \(K_n\) is Hamiltonian, as there exists a Hamiltonian cycle visiting every vertex exactly once and returns to the starting vertex.

Key Concepts

Complete graph

A complete graph, denoted as K_n, is a pivotal structure within the world of graph theory. Imagine a party where every single person shakes hands with every other attendee. That's what a complete graph looks like in a mathematical context—every vertex (representing a person in our analogy) is connected by a unique edge (the handshake) to every other vertex. A key characteristic of a complete graph is its number of edges; it hosts exactly n(n - 1) / 2 edges, where n is the number of vertices.

Understanding the completeness of a graph is essential, especially when solving problems related to traversal and connectivity. It ensures that a journey can always be planned from one vertex to any other, which is one reason why finding a Hamiltonian cycle in a complete graph is a straightforward task—it inherently meets the basic requirement for such a cycle: interconnectedness among all points.

Graph theory

Graph theory is a vast and fascinating field of mathematics that dwells into the properties and interactions of graphs. Defining a graph simply, it consists of vertices (also called nodes or points) connected by edges (also termed as links or lines) that can model virtually any type of relationship. The beauty of graph theory is its versatility; whether it's social networks, computer networks, biological ecosystems, or transportation systems, the possible applications are boundless.

The study of graph theory encompasses various types of graphs, each with specific properties and potential. For example, a graph might be directed or undirected, weighted or unweighted, cyclic or acyclic and may come with its own set of rules and theorems that dictate its underlying structure and behavior. It is these nuances of graph theory that provide the tools to solve complex, real-world problems.

Ore's Theorem

Ore's Theorem is a shining gem within graph theory that provides a sufficient condition for the existence of a Hamiltonian cycle in a graph. It states that if a graph G with n vertices (where n ≥ 3) has the property that for every pair of non-adjacent vertices x and y, the sum of their degrees is at least n, then the graph contains a Hamiltonian cycle. In simpler terms, if you pick any two vertices that don't have an edge between them, their connections to the rest of the graph must be significant enough (at least to the extent of n) to loop a cycle through all vertices.

Applied to a complete graph K_n, each vertex's degree is n - 1. Thus, the sum of the degrees of any pair of distinct vertices is 2(n - 1), which will always be greater than or equal to n, as required by Ore's Theorem. Hence, the existence of a Hamiltonian cycle is guaranteed in a complete graph K_n.

Dirac's Theorem

Dirac's Theorem, similar to Ore's Theorem, provides an invaluable rule to ascertain the presence of a Hamiltonian cycle in a graph. It states that a graph G with n vertices is Hamiltonian if every vertex has a degree of at least n/2. Think of it as a criteria ensuring that each point has enough reaching out or 'connective power' to contribute to a complete cycle that visits every vertex exactly once before returning to the starting point.

When considering a complete graph K_n, Dirac's Theorem becomes rather straightforward to validate. Since this graph type has all its vertices interconnected, the degree of each vertex amounts to n - 1, which is always greater than or equal to n/2 for all values of n ≥ 2. This incontrovertibly proves that a complete graph K_n must contain a Hamiltonian cycle, as it satisfies Dirac's notion of sufficient vertex connectivity.