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Problem 13

# Show that every graph with two or more nodes contains two nodes that have equal degrees.

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The Pigeonhole Principle states that if $$n$$ items are put into $$m$$ containers and $$n > m$$, at least one container must hold more than one item. In a graph with $$n$$ nodes, the maximum degree of a node is $$n-1$$ since a single node can have edges connecting to all the other nodes. If we have $$n$$ nodes and $$n-1$$ possible degrees, then at least two nodes must share the same degree due to the Pigeonhole Principle. In a two-node graph, either the nodes share an edge (both having degree 1) or the nodes have no edges (both having degree 0). In both cases, the two nodes share the same degree. Therefore, every graph with two or more nodes contains two nodes that have equal degrees.
See the step by step solution

## Step 1: Understand the Pigeonhole Principle

The Pigeonhole Principle states that if n items are put into m containers, at least one container must hold more than one item, given that n > m. In the context of our problem, the items are nodes, and the containers are the potential degrees.

## Step 2: Define the maximum degree of a node in a graph

The maximum degree of a node in a graph with n vertices is n-1. This is because, in the worst case, a single node can have edges connecting to all the other nodes in the graph.

## Step 3: Apply the Pigeonhole Principle to the graph

In a graph with n nodes, there are n possible items (each node) and n-1 possible containers (the maximum degree for a node is n-1). According to the Pigeonhole Principle, if we have n nodes, and n-1 possible degrees, then at least two nodes must share the same degree, because there are more nodes than possible degrees.

## Step 4: Address the case when n = 2 (two-node graph)

When there are only two nodes in a graph, either the nodes share an edge (and each have a degree of 1), or the nodes have no edges (and each have a degree of 0). In both cases, we can see that the two nodes share the same degree. Thus, we have shown that every graph with two or more nodes contains two nodes that have equal degrees.

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