Hamiltonian Graph

Duration: 4 min

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  23. UPPSC Live Class Recordings: UPPSC Polytechnic Lecturer 2025 (CS)
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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).

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