Chromatic index χ'(G): minimum colors for proper edge coloring. Vizing's theorem: Δ(G) ≤ χ'(G) ≤ Δ(G)+1. Class 1: χ'=Δ. Class 2: χ'=Δ+1. König (1916): bipartite graphs are always Class 1. χ'(G) = χ(L(G)).
How to Use
Select graph:
- χ': Chromatic index
- Vizing: Class 1 or 2
- König: Bipartite → Class 1
Vizing's Theorem
Every simple graph has χ' = Δ or Δ+1 (Vizing, 1964). Determining which is NP-complete! But many graph families are known: bipartite=Class 1, Petersen=Class 2, planar Δ≥7=Class 1.
Applications
Scheduling: edge coloring models round-robin tournaments. Each round = one color. Minimum rounds = χ'. Network switch scheduling: avoid conflicts on shared links. Timetabling: class-teacher assignments.
Step-by-Step Instructions
- 1Select graph.
- 2Compute Δ.
- 3Determine χ'.
- 4Classify Vizing.
- 5Check König.