Alphabeta Math
DefinitionDefinition: Literature-sourcedProof: Not applicableSession-authored (Fable 5 assisted)audited 2026-08-02
How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

  • Literature-sourced — the exact statement appears in a cited source; only wording and notation differ.
  • AI-adapted — a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
  • AI-generated — a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

Vertex covers, edge covers and the covering numbers τ(G)\tau(G) and ρ(G)\rho(G)

Definition

For a finite simple graph G=(V,E)G=(V,E), a vertex cover is a set C⊆VC\subseteq V such that every edge has an endpoint in CC; its least possible size is τ(G):=min⁡{∣C∣:C is a vertex cover of G}.\tau(G):=\min\{\lvert C\rvert:C\text{ is a vertex cover of }G\}. An edge cover is a set F⊆EF\subseteq E such that every vertex is incident with an edge of FF; if GG has no isolated vertices, its least possible size is ρ(G):=min⁡{∣F∣:F is an edge cover of G}.\rho(G):=\min\{\lvert F\rvert:F\text{ is an edge cover of }G\}. Both extrema exist whenever their displayed families are nonempty, since VV and EE are finite. An isolated vertex prevents every edge cover.

Depends on

Used by

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 25 results over 12 levels. An arrow runs from a result to what uses it, and this result sits at the bottom with a heavier outline. Click the chart to enlarge it.

Sources