Determining if a graph's vertices can be divided into two disjoint sets.
A bipartite graph is a graph whose vertices can be divided into two disjoint and independent sets, U and V, such that every edge connects a vertex in U to one in V. In other words, there are no edges between vertices within the same set. A classic example is a graph of job applicants and available jobs, where an edge represents an applicant being qualified for a job. A key property of bipartite graphs is that they do not contain any odd-length cycles. This property provides the basis for an algorithm to check if a graph is bipartite. We can use either Breadth-First Search (BFS) or Depth-First Search (DFS) for this task. The algorithm involves coloring the vertices with two colors (e.g., red and black). We start by coloring the source vertex with the first color. Then, during traversal, we color all its neighbors with the second color. For each new vertex we visit, we color it with the opposite color of its parent. If at any point we encounter an edge that connects two vertices of the same color, we have found an odd-length cycle, and the graph is not bipartite. If the traversal completes without any such conflicts, the graph is bipartite. This algorithm has many applications, including in matching problems and scheduling.