Complete coloring: proper coloring where every pair of color classes has ≥1 edge between them. ψ(G) = maximum such k = achromatic number. Maximizes colors while ensuring complete inter-class connectivity. Dual of harmonious (≤1 edge per pair).
How to Use
Select graph:
- ψ: Complete coloring #
- Pairs: All adjacent
- Max k: Maximum colors
Bounds
χ(G) ≤ ψ(G) ≤ ψ_s(G) (pseudoachromatic) ≤ ⌊(1+√(1+8m))/2⌋. Upper bound from C(ψ,2) ≤ m. For K_n: ψ = n. For trees: ψ = ⌊(1+√(1+8(n-1)))/2⌋.
Dual of Harmonious
Harmonious: each pair of colors on ≤1 edge (minimizes colors). Complete: each pair on ≥1 edge (maximizes colors). Together they bound how colors can be distributed across edges. Beautiful duality.
Step-by-Step Instructions
- 1Select graph.
- 2Find ψ(G).
- 3Verify all pairs adjacent.
- 4Maximize colors.
- 5Compare with harmonious.