25.1.5.1. Necessary and Sufficient Condition for Complete Matching

Try these first

1. 1.

   Define a bipartite graph.

   Hint

   Think about the two distinct groups in the graph.

2. 2.

   What does matching mean in the context of graphs?

   Hint

   Consider how you could pair up objects without overlap.

3. 3.

   What is a bipartite graph?

   • A graph with one set of vertices
   • A graph where edges connect vertices from two distinct sets
   • A complete graph

   Hint

   Consider the definition and how vertices are arranged.

4. 4.

   True or False: A complete matching always exists in every bipartite graph.

   • True
   • False

   Hint

   Think about conditions necessary for complete matching.

5. 5.

   Given a bipartite graph with subsets U={1, 2, 3} and V={a, b} where U has preferences: 1-a, 1-b, 2-a, 3-a, determine if complete matching is possible.

   Hint

   Use Hall's theorem to check against preferences.

6. 6.

   In a graph representing students to projects, students S={X, Y} can do projects P={1, 2, 3}. X can do 1 & 2, Y can do 2 & 3. Is there complete matching?

   Hint

   Analyze the connections using Hall's theorem conditions.