Alphabeta Math
CorollaryStatement: AI-adaptedProof: AI-generatedSession-authored (Fable 5 assisted)precheck passaudited 2026-08-01
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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.

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Every nonempty hereditary graph class contains the null graph

Statement

Every nonempty hereditary class of finite graphs contains the null graph K0K_0.

Facts & Assumptions

Given: A nonempty hereditary graph class C\mathcal C.

[F1]

Choose G∈CG\in\mathcal C.

[F2]

The induced subgraph G[∅]G[\varnothing] is the null graph K0K_0 (Subgraphs, induced subgraphs and spanning subgraphs, Empty and complete graphs, complete bipartite graphs, and the convention that PnP_n and CnC_n have nn vertices).

[F3]

A hereditary class contains every induced subgraph of each member (Hereditary graph classes).

Proof

technique · direct
1.1

Since G∈CG\in\mathcal C, heredity gives G[∅]∈CG[\varnothing]\in\mathcal C.

F1F3
2.1

Since G[∅]=K0G[\varnothing]=K_0, the null graph belongs to C\mathcal C.

step 1.1F2∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · next 3 levels

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Sources