Problem 18 - Thinking Mathematically

Question

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.

Step-by-Step Solution

Answer

The graph neither has an Euler path nor an Euler circuit.

Step 1: Analyzing the given information

In this step, we note down the given details about the graph. The graph is connected and it has 77 even vertices and four odd vertices.

Step 2: Apply Euler's theorem conditions

Now, let's remember the conditions of Euler's theorem: a graph will have an Euler path, but not an Euler circuit if it is connected and exactly two vertices are of odd degree. The graph will have an Euler circuit if it is connected and all vertices are of even degree.

Step 3: Compare with the given graph

Comparing this with our graph, it is clear that our graph does not satisfy the condition for having an Euler circuit since it has four odd degree vertices. It also does not satisfy the condition for having just an Euler path since there are more than two vertices of odd degree.

Key Concepts

Connected Graph

A connected graph is a type of graph in which there is a path between any pair of vertices. This means that by following the edges of the graph, you can travel from any vertex to any other vertex. Understanding whether a graph is connected is important because it sets the foundation for determining possible Euler paths or circuits. A disconnected graph, on the other hand, has at least one pair of vertices that do not have a path between them. In such cases, it is impossible to find a single Euler path or circuit that traverses every edge exactly once without lifting your pen. In the exercise, the given graph is connected, meaning all vertices are part of the same whole, indicating potential for having Euler paths or circuits depending on other conditions.

Euler's Theorem

Euler's theorem gives us the criteria to determine if a graph has an Euler path or circuit.

• An Euler circuit exists if the graph is connected and all vertices have an even degree. This means every vertex connects to an even number of edges.
• An Euler path (but not a circuit) exists if the graph is connected and exactly two vertices have an odd degree. This means only two vertices have connections to an odd number of edges, with the rest being even.

These statements help identify pathways within a graph that satisfy Euler’s conditions, letting us know if it is possible to traverse every edge exactly once, returning to the starting vertex (Euler circuit), or moving from one vertex to another (Euler path) without returning to the start.

Odd Vertices

In graph theory, an odd vertex is one that has an odd degree, meaning it is connected to an odd number of edges. The degree of a vertex is important because it influences the pathfinding potential in a graph based on Euler's theorem. For a graph to have an Euler path (but not a circuit), exactly two vertices must be odd. These vertices serve as the start and end points for the Euler path. However, if a graph has more than two odd vertices, like in the exercise with four, the graph cannot have an Euler path or circuit. The presence of more than two odd vertices disrupts the balance needed for continuous traversal without lifting a pen.

Even Vertices

An even vertex is characterized by having an even number of edges connecting to it, known as its degree. Even vertices are significant in graph theory as they influence the possibility of forming Euler circuits. In a completely even graph, where all vertices have an even degree, the potential for an Euler circuit is high. This is because you can start at a vertex and return by traversing each edge exactly once. In the given exercise, the 77 even vertices indicate a large proportion of connectivity following Euler's conditions. However, the presence of any odd vertices (specifically four in this case) means the graph can't have an Euler circuit, despite having many even vertices.