Computer Science Engineering (CSE) Exam > Computer Science Engineering (CSE) Questions > Consider the following statements about Bipar...
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Consider the following statements about Bipartite graph:
- The maximum number of edges in a bipartite graph with 10 vertices is 28.
- The maximum number of edges in a bipartite graph with 24 vertices is 143.
- The minimum number of edges in a complete bipartite graph with 10 vertices is 9.
- Every bipartite graph is 2 colorable.
The number of correct statements is _______________.
Correct answer is '2'. Can you explain this answer?
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Most Upvoted Answer
Consider the following statements about Bipartite graph: The maximum n...
Explanation:
Bipartite Graph:
A Bipartite Graph is a graph whose vertices can be divided into two independent sets, U and V such that every edge (u, v) either connects a vertex from U to V or a vertex from V to U.
Maximum Number of Edges:
The maximum number of edges in a bipartite graph with m and n vertices are given by mn.
- The maximum number of edges in a bipartite graph with 10 vertices is 28.
- Here, we can divide the vertices into two sets of 5 vertices each, say U and V.
- The maximum number of edges possible will be the product of the number of vertices in both the sets, i.e., 5 * 5 = 25.
- But, since the edges can only be between the two sets, we can consider only 10 edges (5 * 2).
- Hence, the statement is incorrect.
- The maximum number of edges in a bipartite graph with 24 vertices is 143.
- Here, we can divide the vertices into two sets of 12 vertices each, say U and V.
- The maximum number of edges possible will be the product of the number of vertices in both the sets, i.e., 12 * 12 = 144.
- But, since the edges can only be between the two sets, we can consider only 143 edges (12 * 11).
- Hence, the statement is correct.
Minimum Number of Edges:
The minimum number of edges in a complete bipartite graph with m and n vertices are given by mn.
- The minimum number of edges in a complete bipartite graph with 10 vertices is 9.
- Here, we can divide the vertices into two sets of 5 vertices each, say U and V.
- The minimum number of edges possible will be the product of the number of vertices in both the sets, i.e., 5 * 5 = 25.
- But, since this is a complete bipartite graph, we need to consider all the edges between the two sets, i.e., 10 edges.
- Hence, the statement is correct.
2-Colorable:
Every Bipartite Graph is 2-Colorable, i.e., the vertices of the graph can be colored using 2 colors such that no two adjacent vertices have the same color.
- Every bipartite graph is 2 colorable.
- This statement is correct because, by definition, a bipartite graph can be divided into two sets of vertices, and the edges only exist between these two sets. Hence, we can color the vertices in one set as one color and the vertices in the other set as another color, ensuring that no two adjacent vertices have the same color.
Therefore, the number of correct statements is 2.
Bipartite Graph:
A Bipartite Graph is a graph whose vertices can be divided into two independent sets, U and V such that every edge (u, v) either connects a vertex from U to V or a vertex from V to U.
Maximum Number of Edges:
The maximum number of edges in a bipartite graph with m and n vertices are given by mn.
- The maximum number of edges in a bipartite graph with 10 vertices is 28.
- Here, we can divide the vertices into two sets of 5 vertices each, say U and V.
- The maximum number of edges possible will be the product of the number of vertices in both the sets, i.e., 5 * 5 = 25.
- But, since the edges can only be between the two sets, we can consider only 10 edges (5 * 2).
- Hence, the statement is incorrect.
- The maximum number of edges in a bipartite graph with 24 vertices is 143.
- Here, we can divide the vertices into two sets of 12 vertices each, say U and V.
- The maximum number of edges possible will be the product of the number of vertices in both the sets, i.e., 12 * 12 = 144.
- But, since the edges can only be between the two sets, we can consider only 143 edges (12 * 11).
- Hence, the statement is correct.
Minimum Number of Edges:
The minimum number of edges in a complete bipartite graph with m and n vertices are given by mn.
- The minimum number of edges in a complete bipartite graph with 10 vertices is 9.
- Here, we can divide the vertices into two sets of 5 vertices each, say U and V.
- The minimum number of edges possible will be the product of the number of vertices in both the sets, i.e., 5 * 5 = 25.
- But, since this is a complete bipartite graph, we need to consider all the edges between the two sets, i.e., 10 edges.
- Hence, the statement is correct.
2-Colorable:
Every Bipartite Graph is 2-Colorable, i.e., the vertices of the graph can be colored using 2 colors such that no two adjacent vertices have the same color.
- Every bipartite graph is 2 colorable.
- This statement is correct because, by definition, a bipartite graph can be divided into two sets of vertices, and the edges only exist between these two sets. Hence, we can color the vertices in one set as one color and the vertices in the other set as another color, ensuring that no two adjacent vertices have the same color.
Therefore, the number of correct statements is 2.
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Community Answer
Consider the following statements about Bipartite graph: The maximum n...
In a bipartite graph, the maximum edges will be present when both the partition have equal number of vertices.
That's means with 10 vertices the maximum no. of edges we can have is 25(5×5).
With 24 vertices the maximum no. of edges we can have is 144.
That is 12×12.
Both first and 2nd statement is wrong.
Statement 3 is true, because we have minimum edges in complete graph when one partition has only 1 vertex and the other partition have 9 vertices. So, number of edges is 9.
Statement 4 is true, Because color the one partition with one color and other partition with other color. As there is no edge between the element of a single partition, there will be no problem.
Hence, the correct answer is 2.
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Consider the following statements about Bipartite graph: The maximum number of edges in a bipartite graph with 10 vertices is 28. The maximum number of edges in a bipartite graph with 24 vertices is 143. The minimum number of edges in a complete bipartite graph with 10 vertices is 9. Every bipartite graph is 2 colorable.The number of correct statements is _______________.Correct answer is '2'. Can you explain this answer?
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