Question

If \(G\) is a simple graph with at least two vertices, prove that \(G\) must contain two or more, vertices of the same degree.

Short answer

The Pigeonhole Principle confirms the statement that a simple graph \(G\) with at least two vertices must contain two or more vertices of the same degree. This is because the number of vertices is greater than the number of possible distinct degrees, which ensures at least two vertices share a degree.

Step-by-step solution

Understanding Terms and Definitions

In graph theory, a vertex means a point where two or more lines meet. A 'simple' graph is a graph where no two edges share more than one vertex (so there are no loop edges). The degree of a vertex is the number of edges that connect to it.

Pigeonhole principle

The Pigeonhole principle is a mathematical theory that states: If n items are put into m containers, with n > m, then at least one container must contain more than one item. In the case of this problem, vertices are the items and degrees are the containers.

Relating Pigeonhole Principle to the Graph

In our case, the simple graph \(G\) has \(n\) vertices, which implies there are \(n-1\) different degrees (from 0 to \(n-1\)). This is because a vertex cannot have a degree more than \(n-1\), as it cannot be connected to itself in a simple graph. Therefore, the vertices are the items and the degrees from 0 to \(n-1\) are the containers.

Concluding the Proof

Since the number of vertices \(n\) is greater than the number of distinct possible degrees (\(n-1\)), according to the Pigeonhole Principle, at least two vertices must have the same degree. This proves the statement.

Key concepts

Understanding Simple Graphs

In the world of graph theory, a *simple graph* plays an essential role. It's a type of graph that is quite straightforward, hence the name "simple." A simple graph is a graph where each pair of distinct vertices is connected by at most one edge. Importantly, it does not include loops—no vertex is directly connected to itself. Furthermore, there are no multiple edges directly linking two vertices. Such properties help simplify computations and understanding.

Simple graphs are foundational in studying network structures where loops and multiple connections are not relevant. They have

• vertices, which are the points or nodes of the graph;
• edges, which are the lines connecting the vertices;
• a constraint where every edge connects two different vertices with a maximum of one direct connection between any pair of vertices.

Examples of simple graphs include basic organizational diagrams or social networks where each person is shared with one unique connection.

Exploring the Pigeonhole Principle

The *Pigeonhole Principle* is a simple yet profoundly powerful concept in mathematics, and it is frequently used in various proof scenarios, including graph theory. Essentially, this principle posits that if you have more items than containers, at least one container must hold more than one item. Imagine having 10 apples and only 9 baskets; you'll have to place at least one basket with two apples.

In the context of graphs, this principle provides a fundamental tool. Suppose we have a simple graph with multiple vertices. The degrees of these vertices can fall within a range from 0 to potentially the number of vertices minus one. However, when we have more vertices than possible unique degrees, the principle guarantees some vertices share the same degree.

This principle is key to proving many results in graph theory by offering an intuitive way to tackle problems related to distributing properties among different graph elements.

Understanding Vertex Degree

The *vertex degree* is a central concept when dealing with graphs. It represents the number of edges connected to a given vertex. In simple graphs, this is easy to count—as there are no loops or multiple edges—but it still provides valuable insight into the graph's structure.

Each vertex can have a degree ranging from 0 (if it's isolated) to one less than the total number of vertices (\(n-1\)) because it can't connect to itself. This countable aspect makes it easier to understand and manipulate graphs mathematically.

• Higher vertex degrees indicate a node is more "connected" within the graph.
• Low-degree vertices might suggest isolation or minimal connectivity.

When considering a graph as more than just a mathematical object—say, as a social network—vertex degree can symbolize relationships or interactions, making it a powerful tool in both analyzing and understanding the dynamics within networks.