Bipartite Graphs and Matching: Fundamentals Quiz Quiz

Explore essential concepts in bipartite checking and matching within graphs with this interactive quiz designed to reinforce your understanding of fundamental graph applications. Ideal for learners aiming to grasp graph bipartition, perfect matchings, and related principles in network theory and combinatorics.

1. Identifying Bipartite Graphs

   Which of the following characteristics correctly describes a bipartite graph?

   1. It contains at least one cycle of odd length.
   2. Its edges form a directed cycle.
   3. Its vertex set can be divided into two sets with no edge joining vertices within the same set.
   4. Every vertex connects to every other vertex.

   Explanation: A bipartite graph can be separated into two sets such that every edge connects a vertex from one set to the other, with no connections within a single set. Option B is incorrect because the presence of an odd cycle means the graph cannot be bipartite. Option C describes a complete graph, not necessarily bipartite. Option D refers to directed cycles, which are unrelated to the definition of bipartite graphs.

2. Odd Cycles and Bipartiteness

   If an undirected graph contains an odd-length cycle, what can be concluded about its bipartiteness?

   1. The graph is definitely not bipartite.
   2. The graph is always bipartite.
   3. The graph must be complete.
   4. The graph has parallel edges.

   Explanation: A graph with an odd-length cycle cannot be bipartite because it is impossible to separate its vertices into two sets such that no set contains an edge within itself. Option B is not implied by the presence of an odd cycle. Option C is directly contradicted by the requirement for bipartiteness. Option D refers to multiple edges between two vertices, which is unrelated.

3. Bipartitioning Example

   Given a simple graph where vertices are divided into sets U = {1, 2} and V = {A, B}, and edges are (1, A), (2, B), which property does this graph exhibit?

   1. Bipartite structure
   2. Self-loop
   3. Triangular connection
   4. Complete cycle

   Explanation: The edges only connect vertices between sets U and V, making it a bipartite structure. Option B is incorrect since there is no cycle. Option C is wrong because there are no self-loops (edges from a vertex to itself). Option D, triangular connection, requires three connected vertices forming a cycle, which is not present here.

4. Matching in Bipartite Graphs

   In a bipartite graph, what is a matching?

   1. A set of edges with no shared vertices
   2. A set of all possible paths
   3. A connected component
   4. A collection of cycles

   Explanation: A matching is defined as a set of edges such that no two edges share a common vertex. Option B is unrelated to the definition, as matching does not involve cycles. Option C refers to paths, which are not matchings. Option D confuses matching with connected components, which is a different concept.

5. Perfect Matching Definition

   Which statement best describes a perfect matching in a bipartite graph?

   1. All edges connect every possible pair of vertices.
   2. Each set contains the same number of vertices.
   3. Matching contains only the shortest edges.
   4. Every vertex in both sets is included in exactly one edge of the matching.

   Explanation: In a perfect matching, every vertex is matched to exactly one other vertex with no remainder. Option B refers to a complete bipartite graph, not to perfect matching. Option C is not relevant; edge length does not define matching. Option D describes a necessary condition for a perfect matching but does not summarize what a perfect matching is.

6. Checking Bipartiteness with Coloring

   When performing a bipartite check using coloring, what does it mean if you can color the graph's vertices with two colors without adjacent vertices sharing a color?

   1. The graph is acyclic.
   2. The graph is bipartite.
   3. The graph has parallel edges.
   4. The graph is directed.

   Explanation: If a graph can be colored with two colors such that no adjacent vertices share a color, it is bipartite by definition. Option B (directed) is a different graph property. Option C (acyclic) does not necessarily follow from two-colorability. Option D is unrelated; parallel edges do not affect coloring in this context.

7. Application of Matchings Example

   Suppose in a job assignment scenario, workers are vertices in set U and jobs in set V; an edge connects a worker to a job they can do. What does finding a matching solve in this context?

   1. Counting the number of components
   2. Assigning workers to jobs with no overlap
   3. Finding the shortest path between jobs
   4. Detecting cycles among workers

   Explanation: Finding a matching ensures each worker is assigned to at most one job and vice versa, with no overlaps. Option B relates to shortest paths, which is different. Option C concerns cycles, not matchings. Option D is about connected components, unrelated to the matching problem.

8. Maximum vs. Perfect Matching

   What distinguishes a maximum matching from a perfect matching in a bipartite graph?

   1. A maximum matching must be unique.
   2. Both terms mean the exact same thing.
   3. A perfect matching uses only the shortest edges.
   4. A maximum matching is the largest possible, while a perfect matching matches all vertices.

   Explanation: A maximum matching is as large as possible, but may not match all vertices, while a perfect matching ensures every vertex is paired. Option B is incorrect since maximum matchings need not be unique. Option C incorrectly brings up edge lengths, which are not relevant. Option D is wrong because not every maximum matching is perfect.

9. Testing Bipartiteness with BFS

   If using breadth-first search to check bipartiteness, which feature indicates the graph is not bipartite?

   1. There are no cycles at all.
   2. Every vertex has even degree.
   3. A back edge connects two vertices of the same level.
   4. Each vertex has a self-loop.

   Explanation: A back edge between vertices on the same BFS level reveals an odd-length cycle, which makes the graph non-bipartite. Option B (even degree) does not guarantee bipartiteness. Option C (no cycles) relates more to acyclic graphs, but a graph may be bipartite even with cycles. Option D (self-loop) prevents bipartiteness, but BFS detection focuses on levels and cycles.

10. Real-World Use of Bipartite Graphs

    Which real-world problem can naturally be modeled with bipartite graphs and matching?

    1. Finding the fastest route on a road map
    2. Matching students to available dorm rooms
    3. Calculating the area of geometric shapes
    4. Sorting numbers into order

    Explanation: Assigning students (one set) to rooms (another set) with appropriate conditions can be modeled as a bipartite matching problem. Option B involves shortest path algorithms. Option C is unrelated to graph theory. Option D (sorting numbers) does not involve bipartite graphs or matching.