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Problem 13

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

Expert verified

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.

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Chapter 0

Ramsey's theorem. Let \(G\) be a graph. A clique in \(G\) is a subgraph in which every two nodes are connected by an edge. An anti-clique, also called an independent set, is a subgraph in which every two nodes are not connected by an edge. Show that every graph with \(n\) nodes contains either a clique or an anti-clique with at least \(\frac{1}{2} \log _{2} n\) nodes.

Chapter 0

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Chapter 0

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Chapter 0

If \(A\) has \(a\) elements and \(B\) has \(b\) elements, how many elements are in $A \times B$ ? Explain your answer.

Chapter 0

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