CounterexampleConstruction: AI-adaptedVerification: AI-generatedprecheck pass

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.

The class of regular graphs is not hereditary

Statement refuted

The class of finite regular graphs is hereditary.

Facts & Assumptions

Given: The cycle $C_{4} = v_{0} v_{1} v_{2} v_{3} v_{0}$ .

[F1]

Every vertex of $C_{4}$ has degree two, so $C_{4}$ is regular (Adjacency, incidence, open and closed neighbourhoods, vertex degree, minimum degree and maximum degree, Empty and complete graphs, complete bipartite graphs, and the convention that $P_{n}$ and $C_{n}$ have $n$ vertices).

[F2]

The induced subgraph on ${v_{0}, v_{1}, v_{2}}$ is $P_{3}$ (Subgraphs, induced subgraphs and spanning subgraphs).

[F3]

A hereditary class is closed under induced subgraphs (Hereditary graph classes).

Counterexample

technique · direct

1.1F1

The graph $C_{4}$ is regular.

1.2F2

Its displayed induced $P_{3}$ has degrees $1, 2, 1$ , so it is not regular.

2.1step 1.1step 1.2F3∎

Thus regular graphs are not closed under induced subgraphs and do not form a hereditary class.

Remarks

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

11 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Reinhard Diestel, Graph Theory, Preview Chapter 1 (standard reference, not scraped)
Valerio Boncompagni, On hereditary graph classes defined by forbidding Truemper configurations (PhD thesis, 2018) (standard reference, not scraped)