26.1.2.1. Base Case for Inductive Proof

Try these first

1. 1.

   Define a bipartite graph.

   Hint

   Think about how the vertices are organized in two distinct groups.

2. 2.

   What is Hall's Marriage Theorem?

   Hint

   Consider its application in matching problems.

3. 3.

   What ensures a complete matching exists in a bipartite graph?

   • Only the number of vertices
   • The number of neighbors and vertices must be equal or greater
   • The edges formed randomly

   Hint

   Think about the relationship between vertices in different sets.

4. 4.

   Hall's Marriage Theorem applies to which type of graphs?

   • True
   • False

   Hint

   Recall the definition of a bipartite graph.

5. 5.

   Given a bipartite graph with 6 vertices in V₁ and 8 in V₂, five vertices in V₁ have enough neighbors in V₂ to meet Hall's condition, one does not, and one vertex in V₂ is left unmatched. Can a complete matching still be formed?

   Hint

   Evaluate the matching possibility based on the violating vertex.

6. 6.

   Create an example of a bipartite graph that satisfies Hall's condition and show how to build a complete matching.

   Hint

   Draw the graph to visualize connections.