# Eulerian graphs

Dive into the fascinating world of Eulerian graphs, a key concept within the Further Mathematics curriculum, enriching your understanding of graph theory and its wide-ranging applications. By examining the Eulerian graph definition, you will gain insight into its unique characteristics and how they distinguish it from other graphs. Delve into a variety of practical examples to enhance your grasp of Eulerian graph properties. Throughout this comprehensive guide, learn to identify and solve challenging Eulerian graph problems, as well as explore the differences between Eulerian and Hamiltonian graphs, their key distinctions, and practical applications. Finally, discover Eulerian graph theorems and their real-world implications as you expand your mathematical toolbox and deepen your appreciation for this complex and intriguing subject.

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## Understanding the Eulerian Graph Definition

A crucial concept in Further Mathematics, particularly in Graph Theory, is the Eulerian graph. By definition, a graph is considered Eulerian if it possesses an Eulerian circuit.

An Eulerian circuit is a closed walk through the graph such that it visits each edge exactly once and returns to the starting vertex.

## Characteristics of Eulerian Graphs

Eulerian graphs possess certain distinct characteristics. The famous mathematician, Leonhard Euler, laid the foundations for Eulerian graphs by discovering the necessary criteria for a graph to have an Eulerian circuit. Here are a few essential characteristics:
• Each vertex in the graph has an even degree.
• The graph is connected, meaning there exists a path between any pair of vertices in the graph.
By satisfying these conditions, you can determine whether a graph is Eulerian.

## Exploring Eulerian Graphs Examples

In Further Mathematics, you’ll often encounter problems related to finding or constructing Eulerian circuits. Here’s a step-by-step guide on how to approach these problems:
1. Verify if the graph is connected. If it is not, it cannot be Eulerian.
2. Check the degree of each vertex. If all vertices have an even degree, the graph is Eulerian.
3. To find the Eulerian circuit, start with any vertex and repeatedly move along the edges while marking visited edges. Return to the starting vertex, ensuring that all edges have been visited exactly once.

Example: Suppose you have a graph with edges {(A, B), (A, C), (B, C), (C, D)}. This graph is connected, and the degree of each vertex is: A(2), B(2), C(4), and D(1). Since vertex D has an odd degree, this graph is not Eulerian.

## Difference Between Eulerian and Hamiltonian Graphs

In Graph Theory, Eulerian and Hamiltonian graphs are both essential concepts. However, they have distinct characteristics and applications.A Hamiltonian graph is defined by the existence of a Hamiltonian cycle, which is a closed walk through the graph that visits each vertex exactly once and returns to the starting vertex. Here are some key distinctions between Eulerian and Hamiltonian graphs:
• Eulerian graphs focus on edges, whereas Hamiltonian graphs focus on vertices.
• In Eulerian graphs, each vertex has an even degree; in Hamiltonian graphs, there is no such condition.
• Finding Eulerian circuits have efficient algorithms, while finding Hamiltonian cycles is an NP-complete problem with no known efficient solutions.
Practical applications of Eulerian graphs include finding optimal routes for objects like garbage trucks or postal deliveries, while Hamiltonian graphs can help solve issues related to scheduling, network routing, and resource allocation.

## Eulerian Graph Properties and Theorems

Several theorems and properties related to Eulerian graphs can be applied to real-world scenarios. One of the most significant theorem is the Euler’s theorem, which states that a connected graph possesses an Eulerian circuit if and only if each vertex has an even degree. In real-world applications, Eulerian graphs can be useful in designing efficient routes for vehicles covering certain areas, such as trucks delivering supplies to multiple locations.

By utilizing the Eulerian graph properties, companies can find the most efficient path, which covers all necessary points with minimal traversal of the same edges and minimal fuel consumption. This results in optimized logistics operations and overall cost reduction.

## Eulerian graphs - Key takeaways

• Eulerian graph definition: a graph with an Eulerian circuit, a closed walk that visits each edge exactly once and returns to the starting vertex

• Characteristics of Eulerian graphs: each vertex has an even degree, and the graph is connected

• Difference between Eulerian and Hamiltonian graphs: Eulerian focuses on edges and even vertex degree, while Hamiltonian focuses on vertices and has no specific vertex degree condition

• Finding Eulerian circuits is more efficient than finding Hamiltonian cycles due to algorithm differences

• Eulerian graph theorem: a connected graph possesses an Eulerian circuit if and only if each vertex has an even degree

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Are all eulerian graphs planar?

No, not all Eulerian graphs are planar. An Eulerian graph is one where all vertices have even degree and you can traverse the graph through every edge once. A planar graph is one that can be drawn on a plane without any edges crossing. These are separate concepts and don't always overlap.

How do you draw an Eulerian graph?

To draw an Eulerian graph, create a graph where all vertices have an even degree, meaning that each vertex has an even number of edges connected to it. Ensure that the graph is connected, so that there is a path connecting every pair of vertices. This way, you'll form a graph with an Eulerian circuit, making it Eulerian.

What is the difference between an Eulerian graph and an Eulerian circuit?

An Eulerian graph is a connected graph where every vertex has an even degree, while an Eulerian circuit is a closed path within the graph that traverses each edge exactly once and returns to the starting vertex. Essentially, an Eulerian circuit is a specific type of path within an Eulerian graph.

What makes an Eulerian graph?

An Eulerian graph is one in which all vertices have an even degree, and it is possible to traverse every edge exactly once without lifting your pen, ultimately returning to the starting vertex. This closed path is called an Eulerian circuit.

How do you know if a graph is Eulerian?

A graph is Eulerian if it has a closed trail containing every edge exactly once. To determine if a graph is Eulerian, first ensure that it is connected. Then, check if all vertices have an even degree. If these conditions are met, the graph is Eulerian.

## Test your knowledge with multiple choice flashcards

What is an Eulerian graph?

What are the two necessary criteria for a graph to have an Eulerian circuit?

How can you determine whether a graph is Eulerian?

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