Hamiltonian Paths Basics Quiz

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| Questions: 15 | Updated: Dec 1, 2025
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1) A Hamiltonian path in a graph is a path that:

Explanation

It must visit each vertex exactly once.

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About This Quiz
Hamiltonian Paths Basics Quiz - Quiz

Are you ready to explore how paths move through every vertex in a graph? This quiz walks you through the core ideas behind Hamiltonian paths—what they are, how they behave, and how they differ from cycles. You’ll work with classic examples like complete graphs, trees, and bipartite graphs to understand... see morewhen Hamiltonian paths exist and when they don’t. As you solve each question, you’ll build confidence identifying vertex patterns, checking degree conditions, and applying fundamental theorems like Dirac’s. Get ready to strengthen your understanding of graph traversal, one vertex at a time! see less

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2) Every Hamiltonian cycle is also a Hamiltonian path.

Explanation

A Hamiltonian cycle contains a Hamiltonian path within it.

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3) A Hamiltonian cycle is a Hamiltonian path that additionally:

Explanation

A Hamiltonian cycle starts and ends at the same vertex.

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4) Which of the following graphs always has a Hamiltonian path?

Explanation

Complete graphs contain Hamiltonian paths and cycles.

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5) A graph with a Hamiltonian cycle must be connected.

Explanation

A cycle cannot occur in a disconnected graph.

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6) Which graph definitely does not have a Hamiltonian cycle?

Explanation

A path graph (P₄) contains no cycles at all.

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7) Which graphs contain a Hamiltonian path?

Explanation

Disconnected graphs cannot have Hamiltonian paths.

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8) In a Hamiltonian path, vertices:

Explanation

Vertices must be visited exactly once.

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9) Dirac’s theorem gives a sufficient condition for a Hamiltonian cycle.

Explanation

Dirac’s theorem ensures Hamiltonicity under degree ≥ n/2.

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10) According to Dirac’s theorem, a graph on n vertices is Hamiltonian if:

Explanation

This is Dirac’s condition for Hamiltonian cycles.

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11) Which of the following is true about all Hamiltonian paths?

Explanation

Hamiltonian paths visit all vertices.

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12) Which statements are true?

Explanation

Paths are more general; cycles require extra conditions.

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13) How many Hamiltonian paths does the complete graph Kₙ contain?

Explanation

Any permutation of vertices defines a Hamiltonian path.

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14) A Hamiltonian path exists in every connected graph.

Explanation

Many connected graphs fail to be Hamiltonian.

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15) A graph has 10 vertices, all of degree 1 except one vertex of degree 9. Does it contain a Hamiltonian path?

Explanation

The high-degree vertex can connect all leaf vertices in sequence.

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A Hamiltonian path in a graph is a path that:
Every Hamiltonian cycle is also a Hamiltonian path.
A Hamiltonian cycle is a Hamiltonian path that additionally:
Which of the following graphs always has a Hamiltonian path?
A graph with a Hamiltonian cycle must be connected.
Which graph definitely does not have a Hamiltonian cycle?
Which graphs contain a Hamiltonian path?
In a Hamiltonian path, vertices:
Dirac’s theorem gives a sufficient condition for a Hamiltonian...
According to Dirac’s theorem, a graph on n vertices is Hamiltonian...
Which of the following is true about all Hamiltonian paths?
Which statements are true?
How many Hamiltonian paths does the complete graph Kₙ contain?
A Hamiltonian path exists in every connected graph.
A graph has 10 vertices, all of degree 1 except one vertex of degree...
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