Arboricity arb(G): minimum number of edge-disjoint forests covering all edges. Nash-Williams formula (1964): arb(G) = max_{H⊆G, |V(H)|≥2} ⌈|E(H)|/(|V(H)|-1)⌉. Elegant min-max theorem. arb ≤ degeneracy + 1. Planar: arb ≤ 3.
How to Use
Select graph:
- arb: Arboricity
- Forests: Decomposition
- NW: Nash-Williams
Nash-Williams Theorem
arb(G) = max ⌈|E(H)|/(|V(H)|-1)⌉ over all subgraphs H with ≥2 vertices. Min-max duality: minimum forests = maximum 'density'. Computable in polynomial time via matroid intersection.
Applications
Network orientation: orient edges s.t. max indegree ≤ arb. Data structure lower bounds: arboricity bounds adjacency query complexity. Graph sparsification: arboricity captures edge density locally.
Step-by-Step Instructions
- 1Select graph.
- 2Compute arboricity.
- 3Apply Nash-Williams.
- 4Decompose into forests.
- 5Verify coverage.