Hamiltonian Paths and Hamiltonian Circuits
๐ผ What Is a Hamiltonian Path?
A Hamiltonian Path is a path that:
- Visits every vertex exactly once
- Never repeats a vertex
- Can start and end at different vertices
Think of the vertices as houses in your neighborhood.
A Hamiltonian Path is like planning a walk where you visit every house once, without going back to any house youโve already seen.
Simple, right?
๐ผ What Is a Hamiltonian Circuit?
A Hamiltonian Circuit (or Hamiltonian Cycle) is almost the same thing, but with one small twist:
- You again visit every vertex once
- But this time, you must come back to the starting vertex
So itโs like making a loop around the whole neighborhood:
Start at your home โ visit every house once โ return home.
๐ฏ Hamiltonian = Focus on Vertices
If you studied Euler paths, they were all about edges.
Hamiltonian paths are different:
๐ The goal is to touch every vertex, not every edge.
This one difference completely changes the way we think about the graph.
๐งฉ Diagram 1: Hamiltonian Path Example
Hereโs a simple graph:
A ----- B ----- C
\ /
\ /
DOne Hamiltonian Path could be:
A โ B โ C โ D
Another:
D โ A โ B โ C
Anything that touches all four vertices exactly once is valid.
๐งฉ Diagram 2: Hamiltonian Circuit Example
A
/ \
B --- C
\ /
DA Hamiltonian Circuit here could be:
A โ B โ D โ C โ A
You visit A, B, D, C onceโฆ and then return to A.
Perfect loop!
๐ Real-Life Analogy (Very Easy to Remember)
Think of a Hamiltonian Path as:
๐ฃ๏ธ A sightseeing trip where you want to see every tourist spot exactly once.
Think of a Hamiltonian Circuit as:
๐ A full round trip where you end up back at your starting point after visiting all spots once.
These โtourist spotโ problems appear everywhere in computer science, especially in optimization and scheduling.
โก Why Hamiltonian Paths Are Harder Than They Look
Euler paths have neat mathematical rules based on degrees.
Hamiltonian pathsโฆ do not.
There is no simple shortcut to check whether a graph is Hamiltonian.
You often need to explore many possible paths, which is why problems like:
- Travelling Salesman Problem (TSP)
- Route optimization
are considered difficult.
But conceptually, the idea is still simple:
๐ Visit every vertex exactly once.
๐ Quick Comparison Table
| Feature | Hamiltonian Path | Hamiltonian Circuit |
|---|---|---|
| Visits every vertex? | โ๏ธ Yes | โ๏ธ Yes |
| Repeats vertices? | โ No | โ No |
| Returns to start? | โ Not required | โ๏ธ Must return |
| Focus is onโฆ | Vertices | Vertices |
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