Circular chromatic number χ_c(G): minimum ratio k/d for a (k,d)-coloring. Colors from Z_k, adjacent vertices must differ by ≥ d (mod k). Always χ(G)-1 < χ_c(G) ≤ χ(G). Equals χ(G) iff no 'room' to reduce. Beautiful rational-valued refinement.
How to Use
Select graph:
- χ_c: Circular χ
- k/d: Parameters
- vs χ: Compare
Classic Examples
C_5: χ=3, χ_c=5/2=2.5. C_7: χ=3, χ_c=7/3≈2.33. K_n: χ_c=χ=n. Kneser K(n,k): χ_c=χ=n-2k+2. Odd cycles: χ_c=(2n+1)/n. Reveals finer structure than χ alone.
Theory
χ_c is always rational. Determining χ_c is NP-hard. For planar graphs: χ_c ≤ 4 (4CT). For triangle-free planar: χ_c ≤ 3 (Grötzsch). Active research area connecting combinatorics to topology.
Step-by-Step Instructions
- 1Select graph.
- 2Compute χ.
- 3Find optimal k/d.
- 4Verify (k,d)-coloring.
- 5Compare χ_c vs χ.