A perfect matching in graph theory refers to a situation where every vertex in a graph is connected to exactly one other vertex, forming pairs without any leftover vertices. This means that if a graph has an even number of vertices, a perfect matching can be found if there are enough edges connecting them. In practical terms, perfect matchings can be applied in various fields, such as network design and resource allocation. For example, in a bipartite graph, where two distinct sets of vertices exist, a perfect matching would pair each vertex from one set with a unique vertex from the other set.