The Berge-Tutte theorem is a fundamental result in graph theory that provides a criterion for determining whether a graph can be perfectly matched. A perfect matching is a set of edges that pairs up all vertices in the graph without any overlaps. The theorem states that a graph has a perfect matching if and only if, for every subset of vertices, the number of odd-sized components in the subgraph formed by removing that subset is less than or equal to the number of vertices in the subset. This theorem is named after Claude Berge and W.T. Tutte, who contributed significantly to the field of combinatorial mathematics. The Berge-Tutte theorem is particularly useful in applications such as network design, scheduling, and resource allocation, where matching problems frequently arise. Understanding this theorem helps in solving complex problems involving relationships between pairs of elements in various fields.