Toughness t(G) = min_S |S|/ω(G-S) over all cut sets. Chvátal (1973): Hamiltonian → t ≥ 1. Open conjecture: t ≥ 2 → Hamiltonian. For chordal: t ≥ 1 → Hamiltonian. Fundamental graph resilience parameter.
How to Use
Select graph:
- t: Toughness
- |S|/ω: Ratio
- Ham: t ≥ 1?
Chvátal's Conjecture
Open since 1973! t ≥ 2 → Hamiltonian? Proved for: chordal (t≥1 suffices), planar (t>1 suffices for 4-connected). One of the biggest open problems in Hamiltonian graph theory.
Computing Toughness
NP-hard in general. For special classes: polynomial. Uses max-flow/min-cut algorithms. Related to vertex connectivity: κ(G)/α(G) ≤ t(G). Practical for small/structured graphs.
Step-by-Step Instructions
- 1Select graph.
- 2Find min |S|/ω.
- 3Check t ≥ 1.
- 4Apply Chvátal.
- 5Compare classes.