Here is a complete, clear, and exam-oriented explanation of Hamiltonian Chains (Hamiltonian Paths) — perfect for BCA/MCA/Engineering/Discrete Mathematics exams.
⭐ HAMILTONIAN CHAINS (HAMILTONIAN PATHS) – IN DETAIL
A Hamiltonian Chain is a path in a graph that visits every vertex exactly once.
It is named after Sir William Rowan Hamilton, who studied such paths in the Icosian Game.
Hamiltonian Chains are central in graph theory, optimization, traveling salesman problem (TSP), and network design.
⭐ 1. What Is a Hamiltonian Chain?
A Hamiltonian chain (also called a Hamiltonian path) is:
A path in a graph that visits each vertex exactly once, but does not need to end where it started.
Important points:
✔ Every vertex appears once
✘ Edges cannot repeat because vertices cannot repeat
✘ It is not required to return to the starting vertex
⭐ 2. Hamiltonian Chain vs Hamiltonian Cycle
| Feature | Hamiltonian Chain (Path) | Hamiltonian Cycle |
|---|---|---|
| Visits vertices | All vertices once | All vertices once |
| Return to start | No | Yes |
| Simple? | Yes (no repetition) | Yes |
| Exists when | Very complex to check | Also complex |
⭐ 3. Properties of Hamiltonian Chains
1️⃣ They visit every vertex exactly once.
2️⃣ They are simple paths (no repeated vertices).
3️⃣ Every Hamiltonian cycle contains a Hamiltonian chain.
4️⃣ Deciding if a Hamiltonian chain exists is NP-complete
→ No simple necessary-and-sufficient condition.
5️⃣ A Hamiltonian chain does not need to cover all edges.
6️⃣ Many graphs have multiple Hamiltonian chains.
⭐ 4. Examples
✔ Example 1: Path graph Pₙ
Graph: A—B—C—D
A Hamiltonian Chain is:
A → B → C → D
(or any order without repetition)
✔ Example 2: Cycle graph Cₙ
Graph: A—B—C—D—A
Hamiltonian chains:
A → B → C → D
B → C → D → A
etc.
Cycle has many Hamiltonian paths.
✔ Example 3: Complete graph Kₙ
Every possible vertex ordering is a Hamiltonian chain.
Number of Hamiltonian chains in Kₙ = n!
⭐ 5. Sufficient Conditions for Existence
There is no simple general test, but several useful sufficient conditions:
⭐ 1. Dirac’s Theorem (1952)
If G has (n ≥ 3) vertices and
[
\deg(v) \ge \frac{n}{2} \ \text{for all } v
]
→ G contains a Hamiltonian cycle, hence a chain.
⭐ 2. Ore’s Theorem (1960)
If G has (n ≥ 3) vertices and for every pair of non-adjacent vertices (u, v):
[
\deg(u) + \deg(v) \ge n
]
→ G contains a Hamiltonian cycle, hence a chain.
⭐ 3. Chvátal’s Theorem (more complex)
Provides degree sequence conditions; used in research, not exams.
⭐ 6. Necessary Conditions (but not sufficient)
These conditions must hold but do not guarantee Hamiltonicity:
1️⃣ Graph must be connected
2️⃣ Must not have a vertex of degree 0
3️⃣ Must not have separate components
4️⃣ Must have at least 2 vertices of degree ≥ 2
These help eliminate impossible cases.
⭐ 7. Hamiltonian Chains in Special Graphs
✔ Trees
Every tree has a Hamiltonian chain
(because trees are connected and acyclic).
BUT trees never have Hamiltonian cycles unless they are a simple cycle graph.
✔ Bipartite Graphs
Only those that satisfy matching constraints can be Hamiltonian.
✔ Complete Graphs (Kₙ)
Always have Hamiltonian chains and cycles.
✔ Grid Graphs
Many grid graphs are not Hamiltonian unless constraints are met.
⭐ 8. Algorithms to Find Hamiltonian Chains
Finding Hamiltonian paths is computationally difficult (NP-complete).
But several algorithms exist:
✔ 1. Backtracking
Try all possibilities recursively.
Slow but correct.
✔ 2. Branch and Bound
Prunes paths that can’t lead to a solution.
✔ 3. Dynamic Programming (Held–Karp)
Used for Traveling Salesman Problem.
✔ 4. Heuristics
- Nearest neighbor
- Greedy algorithms
- Genetic algorithms
Not guaranteed to find an optimal path but useful in practice.
⭐ 9. Applications of Hamiltonian Chains
✔ Traveling Salesman Problem (TSP)
Visit each city once → find Hamiltonian path/cycle.
✔ Route optimization
Delivery routes, drone paths.
✔ DNA Sequencing
Genome assembly uses Hamiltonian paths.
✔ AI and Robotics
Path planning.
✔ Computer Games
Knight’s tour problem on chessboard.
✔ Network Design
Fault-tolerant routing and communication.
⭐ 10. Quick Exam-Oriented Summary
✔ Hamiltonian Chain / Path
A path that visits every vertex exactly once.
✔ Hamiltonian Cycle
A cycle visiting every vertex once and returning to start.
✔ Difference from Euler paths
Hamiltonian = vertices
Euler = edges
✔ Key theorems
- Dirac’s theorem
- Ore’s theorem
✔ Properties
- NP-complete problem
- Exists in complete graphs and many connected graphs
- Trees always contain Hamiltonian paths
✔ Applications
TSP, genome sequencing, AI, robotics, routing.