Hamiltonian Path & Cycle Checker

every vertex exactly once

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About Hamiltonian Path & Cycle Checker

A Hamiltonian path/cycle checker using sufficient conditions: Dirac (δ≥n/2 → Hamiltonian cycle), Ore (d(u)+d(v)≥n for non-adjacent → cycle), Chvátal (degree sequence condition). NP-complete in general. Client-side.

Hamiltonian Path & Cycle Checker Features

  • Dirac test
  • Ore test
  • Chvátal
  • NP-complete
  • Common graphs
Hamiltonian cycle: visit every vertex exactly once. NP-complete to decide! But sufficient conditions: Dirac (1952): δ≥n/2. Ore (1960): d(u)+d(v)≥n non-adjacent. Chvátal (1972): degree sequence. These are polynomial tests.

How to Use

Select graph:

  • Dirac: δ≥n/2?
  • Ore: degree sum?
  • Result: Sufficient?

Dirac & Ore

Dirac: minimum degree ≥ n/2 guarantees Hamiltonian cycle (n≥3). Ore generalizes: d(u)+d(v)≥n for every non-adjacent pair. Ore implies Dirac. Both are best-possible in their class.

NP-Completeness

Determining Hamiltonicity is NP-complete (Karp, 1972). No known polynomial algorithm. Even for cubic planar graphs! But sufficient conditions, heuristics, and special graph classes allow practical solutions.

Step-by-Step Instructions

  1. 1Select graph.
  2. 2Check Dirac.
  3. 3Check Ore.
  4. 4Test Chvátal.
  5. 5Determine result.

Hamiltonian Path & Cycle Checker — Frequently Asked Questions

Why is this harder than Euler?+

Eulerian = every edge once: simple parity condition (all even degrees). Hamiltonian = every vertex once: no simple characterization exists. This P vs NP gap is one of the most fundamental in CS.

What graphs are always Hamiltonian?+

K_n (n≥3), K_{n,n}, sufficiently dense graphs (Dirac/Ore), tournament-like digraphs. Petersen graph is NOT Hamiltonian (counterexample). Hypercubes are Hamiltonian.

What's the Chvátal condition?+

Degree sequence d_1≤d_2≤...≤d_n: if for all i<n/2, d_i≤i implies d_{n-i}≥n-i, then Hamiltonian. Strongest degree-based sufficient condition. Subsumes both Dirac and Ore.

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