Pseudoachromatic number ψ_s(G): maximum k for a coloring (properness not required) where every pair of color classes has ≥1 edge. Relaxation of achromatic number. ψ(G) ≤ ψ_s(G) ≤ ⌊(1+√(1+8m))/2⌋. Measures maximum distinguishable color classes.
How to Use
Select graph:
- ψ_s: Pseudoachromatic
- vs ψ: Compare achromatic
- Improper: Allowed
ψ vs ψ_s
ψ requires proper coloring; ψ_s doesn't. So ψ ≤ ψ_s. The gap measures how much properness costs. For many graphs ψ = ψ_s (properness is 'free'). For others: ψ_s > ψ (properness is expensive).
Upper Bound
ψ_s ≤ ⌊(1+√(1+8m))/2⌋ since we need C(ψ_s, 2) ≤ m (each pair needs at least one edge). This is the same upper bound as achromatic, but pseudoachromatic can sometimes achieve it while achromatic cannot.
Step-by-Step Instructions
- 1Select graph.
- 2Compute ψ_s.
- 3Compare with ψ.
- 4Check pair coverage.
- 5Apply upper bound.