Graph Theory (Euler & Hamiltonian)
Hard
14 mins read
Prerequisites
- Basic Set Theory relations
Introduction
Discrete Mathematics computer science systems algorithm design ka mathematical foundation hai. Graph Theory networks mapping analysis (like roads, routers, internet connections) represent karta hai.
Real-Life Analogy
💡 Courier Delivery Boy Route mapping
Traversing roads without repeat steps
| Concept Term | Real-life Analogy Mapping |
|---|---|
| Eulerian Path | Delivery boy traverses EVERY STREET (edge) exactly once. |
| Hamiltonian Path | Delivery boy visits EVERY HOUSE (vertex) exactly once. |
| Graph Node / Vertex | Physical houses or junction checkpoints. |
| Graph Edge | Connecting roads. |
Detailed Concept Explanation
Graph standard representations map V (Vertices) and E (Edges). - **Eulerian Circuit**: Cyclic path starting and ending at same node which visits every edge exactly once. (Possible only when every vertex has EVEN degree). - **Eulerian Path**: Path which visits every edge exactly once. (Possible if exactly 0 or 2 vertices have odd degree). - **Hamiltonian Circuit**: Cyclic path visiting every vertex exactly once.
Visual Diagram
1Mutual ExclusionNo Sharing (1 Resource = 1 Process)
2Hold & WaitResource hold karke doosre ka wait
3No PreemptionZabardasti resource snatching NO
4Circular WaitWaiting chain forms a closed Loop
🚨 Jab ye CHAAR conditions EKSATH lagti hain, tabhi Deadlock hota hai!
Important Point
- Sum of degrees of all vertices equals 2 * total Edges (Handshaking Theorem).
- Finding Hamiltonian path is NP-Complete, whereas Eulerian path can be solved in linear time.
Mnemonic / Memory Trick
E-E | H-V (Even Edges | House Vertices)
Euler = Edges | Hamiltonian = Vertices
Mnemonic helper to associate Euler with edge traversal and Hamiltonian with vertex traversal.
Avoid This Common Mistake
Students Handshaking theorem calculations directed vs undirected graph loops calculation offsets double tracking updates error.
GATE Exam Insights
Degrees of vertices calculations, planar graph planar region configurations (Euler formula: V - E + R = 2), and chromatic numbers are highly scoring points in GATE.
Practice Mini Quiz
Revision Summary (One-Page Notes)
- •Graphs represent connections between entities.
- •Eulerian paths traverse edges; Hamiltonian paths traverse vertices.
- •Planar graph regions satisfy V - E + R = 2.