What is a Hamiltonian Path?

S3-SA5-0393

Grade Level:

Class 10

AI/ML, Data Science, Physics, Economics, Cryptography, Computer Science, Engineering

Definition

What is it?

A Hamiltonian Path in a graph is a path that visits each vertex (or 'point') exactly once. Think of it like a special route you take where you stop at every location on your map, but never visit the same location twice.

Simple Example

Quick Example

Imagine you have a map of 4 chai stalls in your neighbourhood: A, B, C, D. You want to visit each stall exactly once to taste their chai. A Hamiltonian Path would be a route like A -> B -> D -> C. You started at A, visited B, then D, then C, and you visited all 4 stalls without going to any stall more than once.

Worked Example

Step-by-Step

Let's find a Hamiltonian Path in a simple graph with 4 vertices: V1, V2, V3, V4. The connections (edges) are: V1-V2, V1-V3, V2-V3, V2-V4, V3-V4.

---Step 1: Start at an arbitrary vertex, say V1. Our current path is [V1].

---Step 2: From V1, we can go to V2 or V3. Let's try V2. Our path is [V1, V2]. We have visited 2 vertices.

---Step 3: From V2, we can go to V1 (already visited), V3, or V4. We must visit unvisited vertices. Let's try V3. Our path is [V1, V2, V3]. We have visited 3 vertices.

---Step 4: From V3, we can go to V1 (visited), V2 (visited), or V4. We must visit unvisited vertices. Let's go to V4. Our path is [V1, V2, V3, V4].

---Step 5: We have visited all 4 vertices (V1, V2, V3, V4) exactly once. This is a Hamiltonian Path.

ANSWER: One Hamiltonian Path is V1 -> V2 -> V3 -> V4.

Why It Matters

Hamiltonian Paths are super important in Computer Science and Data Science for solving complex routing problems, like finding the most efficient delivery route for food or packages. They are used by engineers to design circuits and by logistics companies to plan travel for goods, saving time and money.

Common Mistakes

MISTAKE: Visiting a vertex more than once in the path. | CORRECTION: A Hamiltonian Path must visit each vertex *exactly* once. If you revisit, it's not a Hamiltonian Path.

MISTAKE: Not visiting all vertices in the graph. | CORRECTION: The path must include *every* vertex in the graph. If even one vertex is missed, it's not a Hamiltonian Path.

MISTAKE: Confusing a Hamiltonian Path with an Eulerian Path. | CORRECTION: A Hamiltonian Path visits every *vertex* exactly once. An Eulerian Path visits every *edge* exactly once.

Practice Questions

Try It Yourself

QUESTION: A graph has vertices P, Q, R, S and edges P-Q, Q-R, R-S, S-P. Is P -> Q -> R -> S a Hamiltonian Path? | ANSWER: Yes

QUESTION: For a graph with vertices A, B, C, D and edges A-B, B-C, C-D. Can A -> B -> C -> D be a Hamiltonian Path? Why or why not? | ANSWER: Yes, because it visits every vertex (A, B, C, D) exactly once.

QUESTION: Draw a graph with 5 vertices (V1, V2, V3, V4, V5) and edges (V1-V2, V1-V3, V2-V4, V3-V4, V4-V5). Find one Hamiltonian Path. | ANSWER: V1 -> V2 -> V4 -> V3 -> V5 (Many other solutions are possible, e.g., V1 -> V3 -> V4 -> V2 -> V5)

MCQ

Quick Quiz

Which of the following describes a Hamiltonian Path?

A path that visits every edge exactly once.

A path that visits every vertex exactly once.

A path that revisits the starting vertex.

A path that visits some vertices multiple times.

The Correct Answer Is:

A Hamiltonian Path is defined as a path that visits every vertex in the graph exactly once. Option A describes an Eulerian Path, while C and D describe paths that are not Hamiltonian.

Real World Connection

In the Real World

Imagine a delivery person for Swiggy or Zomato who needs to deliver food to 5 different houses in a neighbourhood. Finding the best route that visits each house exactly once and takes the shortest time is a real-world application of finding a Hamiltonian Path. This helps companies like Delhivery or Ecom Express plan efficient logistics.

Key Vocabulary

Key Terms

GRAPH: A collection of points (vertices) and lines (edges) connecting them. | VERTEX: A point or node in a graph. | EDGE: A line connecting two vertices in a graph. | PATH: A sequence of distinct vertices in a graph connected by edges. | EXACTLY ONCE: Visiting something one and only one time.

What's Next

What to Learn Next

Great job understanding Hamiltonian Paths! Next, you can explore 'Hamiltonian Cycles,' which are Hamiltonian Paths that start and end at the same vertex, forming a closed loop. This will deepen your understanding of graph theory and its practical uses.