Hamiltonian Graph

Duration: 4 min

This video lesson is available to enrolled students.

Return to /learn/MPPSC-ASSISTANT-PROFESSOR-COMPUTER-SCIENCE-PAPER-2/unit-i-discrete-mathematical-structure/graph-theory/euler-and-hamiltonian-graphs-definitions-conditions-and-examples/asset-hamiltonian-graph after enrolling

Inside: a video lesson and guided study material.

Module outline

  1. About Course: About the Course
  2. Paper 1 | General Studies: State Wise GK
  3. Paper 1 | Information and Communication Technology: Role of Internet and Web Services, Security in Mobile and Wireless Computing
  4. Paper 1 | Current Affairs: GK- Current Affairs
  5. Paper 2 | Discrete Mathematical Structure: Digital Systems & Boolean Basics, Logic Gates & Hardware, Boolean Expression, Boolean Minimization, Set Theory, Relations, Functions, Graph Theory, Group Theory, Propositional and Predicate Logic, Probability
  6. Paper 2 | Digital Computer Organization And Architecture: Number System, Number Representation, Sequential Circuits, Combinational Circuit, Basics of COA, Floating Point Rep, Cache Memory Organization, Input Output Organisation, Pipelining, Instr Formats & Modes, Register Transfer and Microoperations, Control Unit Design, Programming the Basic Computer
  7. Paper 2 | C Programming: C Fundamentals, Control Flow, Functions, Arrays & Pointers, Storage Classes, Structures & Enums, DMA, Macros, Scoping & File Handling
  8. Paper 2 | Programming Languages & Object-Oriented Programming Concepts: Java, OOPS with C++, Java Fundamentals, Array, String, IO & References
  9. Paper 2 | Python: Introduction to Python, Tokens, Variables, Data Types, I/O, Operators & Expressions, Control Flow & Looping, Collections & Strings, Functions & Modules, File Handling & Data Analysis
  10. Paper 2 | Data Structure And Algorithms: Introduction to DS, Array, Stack, Queue, Linked List, Tree, Graphs, Hashing, Algorithm Analysis, Time Complexity Analysis, Sorting Algorithms, Greedy Algorithms, Minimum Spanning Trees, Advanced Algorithms
  11. Paper 2 | Operating System: Introduction to OS, Process Management, CPU Scheduling, Process Synchronization, Deadlock, Threads & Process Creation, Memory Management, Virtual Memory, Disc Scheduling, File Management, Unix OS
  12. Paper 2 | Theory Of Computation: Introduction to TOC, Deterministic FA (DFA), Non-Deterministic FA, Regular Expressions, Grammar, Regular Language Properties, Moore & Mealy Machines, Pushdown Automata & CFG, Turing Machines, Complexity Theory
  13. Paper 2 | Compiler Design: Intro to Compilers, Lexical Analysis, Grammar & CFG, Syntax Analysis: Top-Down, Syntax Analysis: Bottom-Up, Semantic Analysis & SDT, Intermediate Code Gen, Code Optimization, Run Time Environment
  14. Paper 2 | Database Management System: Basics of DBMS, ER Diagram, Relational Model & Functional Dependencies, Keys & Integrity Constraints, Normalization (1NF - BCNF), Decomposition Properties & 4NF, Relational Algebra, SQL, Transaction Management, Concurrency Control, Database Recovery, Database Security & Authorization, Enhanced Data Models
  15. Paper 2 | Computer Network And Information Security: Introduction to CN, Data Communication, Application Layer, Hardware basics, Cyber Threats and Malware Attacks, Network Security, DLL: Access Control, DLL: Flow Control, DLL: Error Control, DLL: Framing, Data Link Layer - Ethernet, Net Layer: IPv4 & Proto, Net Layer: IP Addressing, Net Layer:Routing Protocol, Transport Layer Services, TL: Congestion & UDP
  16. Paper 2 | Software Engineering: Fundamentals of Software Engineering, Software Requirements and Quality Assurance, Software Design, Estimation and Metrics, Software Testing, Software Maintenance and Configuration Management
  17. Paper 2 | Current Trends & Technologies: Emerging Tech Trends, Approaches to AI, Search Algorithms, Game Playing, Knowledge Representation, Planning, Multi Agent Systems, Fuzzy Sets, Natural Language Processing, Artificial Neural Networks, Genetic Algorithms, Artificial Intelligence, Data Warehousing & Mining, Electronic Data Interchange and E-Commerce, Terms & Abbreviations
  18. Paper 2 | Full Length Mock Tests:
AI summary & chapters

AI Summary

An AI-generated summary of this video lecture.

This lecture introduces the concept of Hamiltonian Graphs in graph theory. The instructor begins by defining a Hamiltonian circuit as a closed walk that traverses every vertex of a connected graph exactly once, except for the starting vertex which is visited at the end. He explicitly states on the slide that finding whether a graph is Hamiltonian or not is an NP-Complete problem. To illustrate, he draws a pentagon-shaped graph with vertices labeled a, b, c, d, and e. He traces a path in blue ink: a to b, b to c, c to d, d to e, and e back to a, demonstrating a valid Hamiltonian circuit. The lecture then shifts to sufficient conditions for a graph to be Hamiltonian, presenting a specific question from the NET-DEC-2018 exam. This question lists three mathematical conditions involving vertex degrees and edge counts that guarantee a graph is Hamiltonian. The instructor concludes by analyzing a set of four different graphs in a table, attempting to classify them as Eulerian or Hamiltonian based on the properties discussed.

Chapters

  1. 0:00 – 2:00 00:00-02:00

    The instructor defines a Hamiltonian circuit using on-screen text: "A Hamiltonian circuit in a connected graph is defined as a closed walk that traverses every vertex of G exactly once." He emphasizes the starting vertex is visited twice. He draws a graph with five vertices labeled a, b, c, d, e. He traces a path in blue ink connecting a-b-c-d-e-a, visually confirming it as a Hamiltonian circuit. He also writes "Finding weather a graph is Hamiltonian or not is a NPC problem" on the slide, highlighting the computational difficulty.

  2. 2:00 – 4:10 02:00-04:10

    The instructor presents a slide with a multiple-choice question from NET-DEC-2018. The question asks for sufficient conditions for a graph G with n>=3 vertices to be Hamiltonian. Three conditions are listed: (i) deg(v) >= n/3, (ii) deg(v) + deg(w) >= n for non-adjacent v and w, and (iii) |E(G)| >= 1/3(n-1)(n-2) + 2. He then shows a table with four graphs to classify. He analyzes a square graph with an internal point, tracing paths to check for Hamiltonian properties. He writes "E -> NP" and "H -> V" on the board, likely referring to complexity classes. He discusses the conditions, noting that condition (ii) is Ore's Theorem, which is a key sufficient condition for Hamiltonicity.

The video progresses from the basic definition of Hamiltonian graphs to their computational complexity and sufficient conditions. The instructor uses a pentagon graph to visually demonstrate a Hamiltonian circuit, tracing the path a-b-c-d-e-a. He then introduces theoretical criteria, specifically citing a NET-DEC-2018 question that lists three conditions involving vertex degrees and edge counts. These conditions are variations of Dirac's and Ore's theorems. The lecture concludes with a practical exercise where the instructor analyzes a table of four graphs, attempting to classify them as Eulerian or Hamiltonian. This practical application reinforces the theoretical conditions discussed earlier, helping students understand how to apply the theorems to specific graph structures. He also contrasts the complexity of Eulerian paths (P) with Hamiltonian paths (NP-Complete).

Loading lesson…