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For which values of m and n does the complete bipartite graph km, n have a Hamilton circuit?
- a)m ≠ n, m, n ≥ 2
- b)m ≠ n, m, n ≥ 3
- c)m = n, m, n ≥ 2
- d)m = 2, m, n ≥ 3
Correct answer is option 'C'. Can you explain this answer?
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For which values of m and n does the complete bipartite graph km, n ha...
For the complete bipartite graph $K_{m,n}$ to have a Hamilton circuit, we need every vertex to have degree at least 2.
The degree of each vertex in the left part of the graph is $n$, since it is connected to every vertex in the right part. Similarly, the degree of each vertex in the right part is $m$.
Therefore, we need $m \geq 2$ and $n \geq 2$ for every vertex to have degree at least 2.
If $m=1$ or $n=1$, then one of the parts of the graph only has one vertex, and thus it cannot have a Hamilton circuit.
So the answer is: $m \geq 2$ and $n \geq 2$.
The degree of each vertex in the left part of the graph is $n$, since it is connected to every vertex in the right part. Similarly, the degree of each vertex in the right part is $m$.
Therefore, we need $m \geq 2$ and $n \geq 2$ for every vertex to have degree at least 2.
If $m=1$ or $n=1$, then one of the parts of the graph only has one vertex, and thus it cannot have a Hamilton circuit.
So the answer is: $m \geq 2$ and $n \geq 2$.
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For which values of m and n does the complete bipartite graph km, n ha...
Concept:
Complete bipartite graph : In this graph, vertices are partitioned into two sets, such that no vertices are connected in the same set and all vertices of one set are connected to all other vertices of different set. All possible pairs of edges are considered except edges in the same set.
Complete bipartite graph : In this graph, vertices are partitioned into two sets, such that no vertices are connected in the same set and all vertices of one set are connected to all other vertices of different set. All possible pairs of edges are considered except edges in the same set.
Explanation:
Hamiltonian circuit: In this circuit, every vertex is visited only once. It must start and end at the same vertex.
In complete bipartite graph Km, n, when m = n, then in that case, it has a Hamiltonian circuit.
Consider m = n = 3.
Hamiltonian circuit possible in this is: {1, 4, 2, 6, 3, 5, 1}
Hamiltonian circuit: In this circuit, every vertex is visited only once. It must start and end at the same vertex.
In complete bipartite graph Km, n, when m = n, then in that case, it has a Hamiltonian circuit.
Consider m = n = 3.
Hamiltonian circuit possible in this is: {1, 4, 2, 6, 3, 5, 1}
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For which values of m and n does the complete bipartite graph km, n have a Hamilton circuit?a)m ≠ n, m, n ≥ 2b)m ≠ n, m, n ≥ 3c)m = n, m, n ≥ 2d)m = 2, m, n ≥ 3Correct answer is option 'C'. Can you explain this answer?
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For which values of m and n does the complete bipartite graph km, n have a Hamilton circuit?a)m ≠ n, m, n ≥ 2b)m ≠ n, m, n ≥ 3c)m = n, m, n ≥ 2d)m = 2, m, n ≥ 3Correct answer is option 'C'. Can you explain this answer? for Civil Engineering (CE) 2025 is part of Civil Engineering (CE) preparation. The Question and answers have been prepared according to the Civil Engineering (CE) exam syllabus. Information about For which values of m and n does the complete bipartite graph km, n have a Hamilton circuit?a)m ≠ n, m, n ≥ 2b)m ≠ n, m, n ≥ 3c)m = n, m, n ≥ 2d)m = 2, m, n ≥ 3Correct answer is option 'C'. Can you explain this answer? covers all topics & solutions for Civil Engineering (CE) 2025 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for For which values of m and n does the complete bipartite graph km, n have a Hamilton circuit?a)m ≠ n, m, n ≥ 2b)m ≠ n, m, n ≥ 3c)m = n, m, n ≥ 2d)m = 2, m, n ≥ 3Correct answer is option 'C'. Can you explain this answer?.
For which values of m and n does the complete bipartite graph km, n have a Hamilton circuit?a)m ≠ n, m, n ≥ 2b)m ≠ n, m, n ≥ 3c)m = n, m, n ≥ 2d)m = 2, m, n ≥ 3Correct answer is option 'C'. Can you explain this answer? for Civil Engineering (CE) 2025 is part of Civil Engineering (CE) preparation. The Question and answers have been prepared according to the Civil Engineering (CE) exam syllabus. Information about For which values of m and n does the complete bipartite graph km, n have a Hamilton circuit?a)m ≠ n, m, n ≥ 2b)m ≠ n, m, n ≥ 3c)m = n, m, n ≥ 2d)m = 2, m, n ≥ 3Correct answer is option 'C'. Can you explain this answer? covers all topics & solutions for Civil Engineering (CE) 2025 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for For which values of m and n does the complete bipartite graph km, n have a Hamilton circuit?a)m ≠ n, m, n ≥ 2b)m ≠ n, m, n ≥ 3c)m = n, m, n ≥ 2d)m = 2, m, n ≥ 3Correct answer is option 'C'. Can you explain this answer?.
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