Independence number α(G): max set of pairwise non-adjacent vertices. α(K_n)=1, α(K_{m,n})=max(m,n), α(C_n)=⌊n/2⌋. By complement: α(G)=ω(Ḡ). Ramsey: R(s,t) guarantees α≥s or ω≥t in any n-vertex graph.
How to Use
Select graph:
- α(G): Independence number
- Set: Example max set
- Complement: ω(Ḡ)
Applications
Scheduling: independent tasks can run simultaneously. Network: non-interfering transmissions. Coding theory: independent codewords in confusion graph. Social: max group with no conflicts. Facility placement: no two too close.
Bounds
α(G) ≥ n/(Δ+1) (greedy). α(G) ≥ Σ 1/(d(v)+1) (Turán-type). For triangle-free: α ≥ Ω(√(n log n)) (Ramsey). α·χ ≥ n always. Perfect graphs: α·χ = n for optimal coloring.
Step-by-Step Instructions
- 1Select graph.
- 2Compute α(G).
- 3See independent set.
- 4Check complement.
- 5Compare bounds.