Fractional chromatic number χ_f(G): LP relaxation of integer chromatic number. ω(G) ≤ χ_f(G) ≤ χ(G). Also χ_f = n/α. For perfect graphs: χ_f=χ=ω. For odd cycles: χ_f(C_{2k+1})=2+1/k (between 2 and 3).
How to Use
Select graph:
- χ_f: Fractional chromatic
- Bounds: ω ≤ χ_f ≤ χ
- Formula: n/α
LP Relaxation
Integer coloring: assign each vertex exactly one color. Fractional: assign weight to independent sets covering each vertex with total weight 1. Minimize total weight. This LP has rational optimal value.
Kneser Graphs
Kneser graph K(n,k): vertices are k-subsets of {1,...,n}, adjacent iff disjoint. χ(K(n,k))=n-2k+2 (Lovász, 1978). χ_f(K(n,k))=n/k. The gap χ-χ_f can be arbitrarily large!
Step-by-Step Instructions
- 1Select graph.
- 2Compute χ_f.
- 3Compare ω,χ_f,χ.
- 4Check n/α.
- 5Identify gap.