Graph genus γ(G): minimum genus of an orientable surface for crossing-free embedding. γ=0: planar (sphere). γ=1: toroidal (torus). Euler: n-m+f = 2-2γ. Ringel-Youngs (1968): complete solution for K_n. NP-hard for general graphs.
How to Use
Select graph:
- γ: Genus value
- Surface: Embedding
- Euler: n-m+f=2-2γ
Ringel-Youngs Theorem
γ(K_n) = ⌈(n-3)(n-4)/12⌉ for n≥3. Solved the Heawood conjecture! 12-year effort (1954-1968) with many collaborators. One of the great achievements of topological graph theory.
Applications
VLSI design: embedding circuits on surfaces. Map coloring: Heawood formula χ ≤ ⌊(7+√(1+48γ))/2⌋. Network routing on surfaces. Computational topology: surface classification.
Step-by-Step Instructions
- 1Select graph.
- 2Compute genus.
- 3Apply Euler.
- 4Check Heawood.
- 5Classify surface.