Problem Set 9 Solutions

Solution. In a finite connected graph with n ≥ 2 vertices, the domain for the vertex degrees is the set {1, 2, . . . , n − 1} since each vertex can be adjacent to at most all of the remaining n−1 vertices and the existence of a degree 0 vertex would violate the assumption that the graph be connected. Therefore, treating the n vertices as the pigeons and the n−1 possible degrees as the pigeonholes, the pigeonhole principle implies that there must exist a pair of vertices with the same degree.