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Chapter 14: Problem 16

In Exercises 15-18, determine the number of Hamilton circuits in a complete graph with the given number of vertices. 4

Short Answer

Expert verified
There are 3 Hamilton circuits in a complete graph with 4 vertices.

Step by step solution

01

Understanding the concept of Hamilton Circuits and Complete Graph

A Hamilton circuit is a path in a graph which visits each vertex exactly once and then returns to the starting vertex. A complete graph is a simple undirected graph in which every pair of distinct vertices is connected by a unique edge.
02

Recognizing the pattern

The number of Hamilton circuits in a complete graph with \(n\) vertices is \((n-1)!\), but this counts each circuit twice since each circuit can be traversed in a clockwise and counterclockwise manner. Therefore, the actual number of circuits is \((n-1)!/2\).
03

Applying the formula

The formula is \(\frac{(n-1)!}{2}\). Here, \(n\) is 4, the number of vertices in the graph. Substituting 4 into the formula gives \(\frac{(4-1)!}{2}\), which simplifies to \(\frac{3!}{2}\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Complete Graph
A complete graph is a special type of graph in graph theory where every pair of distinct vertices is connected by a unique edge. This means that each vertex is directly linked to every other vertex in the graph. Complete graphs are often denoted as \(K_n\), where \(n\) represents the number of vertices in the graph.
  • For example, a complete graph with 3 vertices, \(K_3\), forms a triangle. Each vertex connects to the other two.
  • In a complete graph with 4 vertices, \(K_4\), the graph looks like a quadrilateral where each corner is connected to every other corner.
Complete graphs are important in many areas of mathematics and computer science, particularly when exploring problems involving connectivity and traversal. They provide a fundamental structure on which other more complex systems can be analyzed.
Graph Theory
Graph theory is a branch of mathematics focused on the study of graphs, which are structures used to model pairwise relations between objects. A graph is made up of vertices (sometimes called nodes) and edges (lines that connect pairs of vertices).
  • Graphs can be used to represent many real-world systems, such as social networks, road maps, and communication networks.
  • Key concepts in graph theory include paths, circuits, connectivity, and cycles.
Graph theory is a foundational element in various fields including computer science, biology, and operations research, providing algorithms and methods to solve complex problems like shortest path or network flow.
Combinatorics
Combinatorics is a field of mathematics concerning the counting, arrangement, and combination of objects. It's often used to determine the number of possible configurations in a set. In the context of Hamilton circuits in a complete graph, combinatorics helps us to figure out the number of different ways to visit each vertex once.
  • Using permutations, we can find the distinct orders to arrange \(n\) vertices in a path, accounting for symmetrical duplications.
  • The formula \((n-1)!/2\) emerges from combinatorial reasoning to accurately count the Hamilton circuits without repetition.
By applying combinatorial techniques, we can effectively solve complex counting problems, underpinning many critical algorithms in computer science.
Vertices
Vertices are the fundamental units or 'points' in a graph where edges meet. In a graph, each vertex is connected to other vertices through edges. The number of vertices gives us an insight into the size and complexity of a graph.
  • Vertices can represent various objects depending on the context, like cities in a network, users in a social network, or tasks in a project plan.
  • In a complete graph, every vertex is connected to all other vertices, emphasizing the importance of each vertex in maintaining the overall connectivity of the graph.
Understanding vertices is crucial for analyzing graph structures and understanding how changes to one part of the graph affect the entire network.

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Most popular questions from this chapter

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In Exercises 15-18, determine the number of Hamilton circuits in a complete graph with the given number of vertices. 13

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