Introduction to the Theory of Computation, 3rd Edition

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

Proof. Assume no self-edges (otherwise exists counterexample \((\{0,1\},\{\{1,1\}\})\)) or multiple edges (otherwise counterexample \((\{1,2,3\},\{\{1,2\},\{2,3\},\{2,3\}\})\)). If there is a node with no neighbors, remove it. Now the degree of each node is at least \(1\). Now there are \(m\) nodes, whose degrees can only be from \(\{1,\ldots,m-1\}\). By the pigeonhole principle, two nodes have the same degree.