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Chapter 14: Problem 51

How can you look at a graph and determine if it has a Hamilton circuit?

Short Answer

Expert verified
To determine if a graph has a Hamilton circuit: ensure that the graph is connected, verify that all vertex degrees are at least two, then attempt to traverse the graph, visiting each vertex once without repetition and returning to the start point.

Step by step solution

01

Check for connectedness

Firstly, check if the graph is connected. This means every pair of vertices in the graph should be connected directly or indirectly. This can be done by trying to visit every vertex in the graph from any given point.
02

Check for Degree

Next, ensure that the degree of each vertex in the graph is two or more. The degree of a vertex can be found by counting the number of edges connected to that vertex. A Hamilton circuit cannot exist in a graph where any vertex has a degree of one because every vertex on a Hamilton circuit has two edges connected to it.
03

Keep track of visited vertices

Try traversing the graph in a manner that returns to the starting vertex after visiting each vertex exactly once. It's essential to keep track of vertices visited to ensure there's no repeat.
04

Conclusion

If all these conditions are met, it can be concluded that the graph has a Hamilton circuit. If even one condition is not satisfied, then the graph does not have a Hamilton circuit.

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Most popular questions from this chapter

In Exercises 13-18, a connected graph is described. Determine whether the graph has an Euler path (but not an Euler circuit), an Euler circuit, or neither an Euler path nor an Euler circuit. Explain your answer. The graph has 78 even vertices and two odd vertices.

In your own words, briefly describe how to find the minimum spanning tree using Kruskal's Algorithm.

In Exercises 5-6, draw two equivalent graphs for each description. The vertices are \(A, B, C\), and \(D\). The edges are \(A D, B C, D C\), \(B B\), and \(D B\).

In Exercises 49-52, draw a graph with the given characteristics. The graph has four odd vertices and at least one loop.

In Exercises 11-16, a graph with no loops or more than one edge between any two vertices is described. Which one of the following applies to the description? i. The described graph is a tree. ii. The described graph is not a tree. iii. The described graph may or may not be a tree. The graph has nine vertices and six edges.
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