Harmonious chromatic number h(G): minimum proper coloring where every pair of adjacent color classes has at most one edge between them. Each edge gets a unique 'color pair'. Lower bound: h ≥ ⌈(1+√(1+8m))/2⌉ (need C(h,2) ≥ m).
How to Use
Select graph:
- h: Harmonious χ
- Pairs: Edge colors
- Bound: Lower
Bounds
h(G) ≥ ⌈(1+√(1+8m))/2⌉ since we need C(h,2) ≥ m distinct edge color-pairs. For trees: h = ⌈(1+√(1+8(n-1)))/2⌉. For complete graphs: h = n. Tight for many graph families.
Theory
Related to line graph coloring in disguise. Harmonious is much harder than proper coloring. Even for trees, determining exact h is non-trivial. Upper bounds: h ≤ 2χ'(G) + 1 where χ' is edge chromatic number.
Step-by-Step Instructions
- 1Select graph.
- 2Compute h(G).
- 3Check pair uniqueness.
- 4Apply lower bound.
- 5Compare with χ.