Binding number bind(G) = min |N(S)|/|S| over S with N(S) ≠ V. Woodall: bind ≥ 3/2 → Hamiltonian. bind ≥ 1 → perfect matching (n even). Measures how well neighborhoods expand. Higher binding = more connected structure.
How to Use
Select graph:
- bind: Binding #
- |N(S)|/|S|: Expansion
- Ham: ≥ 3/2?
Neighborhood Expansion
For every subset S: N(S) should be large relative to S. bind(G) captures the worst-case expansion. Related to Hall's theorem and matching theory. Expander-like behavior for high binding.
Key Results
Woodall (1973): bind ≥ 3/2 → Hamiltonian. bind ≥ 1 → perfect matching (even n). bind ≥ (n-1)/(n-δ) where δ = min degree. K_n: bind = n-1. Stars: bind = 1/(n-1).
Step-by-Step Instructions
- 1Select graph.
- 2Find min |N(S)|/|S|.
- 3Check bind ≥ 3/2.
- 4Apply Woodall.
- 5Compare matching.