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Chapter 14: Problem 62

What is an Euler circuit?

Short Answer

Expert verified
An Eulerian circuit is a circuit in a graph that visits every edge exactly once and starts and ends on the same vertex. An Eulerian circuit exists in a graph if and only if the graph is connected and each vertex has an even degree.

Step by step solution

01

Definition

In the mathematical field of graph theory, an Eulerian trail (or Eulerian path) is a trail in a graph which visits every edge exactly once. Similarly, an Eulerian circuit or Eulerian cycle is an Eulerian trail which starts and ends on the same vertex.
02

Conditions

Euler's theorem states that a graph has an Eulerian cycle if and only if it is connected and every vertex has an even degree.
03

Example

For instance, consider a graph with 5 vertices connected in the shape of a star. Here, every vertex has a degree of 2 which is even and the graph is connected. Therefore, this graph has an Eulerian cycle.

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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 80 even vertices and no odd vertices.

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 77 even vertices and four odd vertices.

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

Give an example of a tree with six vertices whose degrees are \(1,1,2,2,2\), and \(2 .\)

Make Sense? In Exercises 60-63, determine whether each statement makes sense or does not make sense, and explain your reasoning. I'm amazed by the power of my new computer, so this evening I plan to use it to find an optimal Hamilton circuit for a complete, weighted graph with 20 vertices.
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