Alphabeta Math
CounterexampleConstruction: AI-adaptedVerification: AI-generatedprecheck 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.

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 C4=v0v1v2v3v0.

[F1]

Every vertex of C4 has degree two, so C4 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 Pn and Cn have n vertices).

[F2]

The induced subgraph on {v0,v1,v2} is P3 (Subgraphs, induced subgraphs and spanning subgraphs).

[F3]

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

Counterexample

technique · direct
1.1

The graph C4 is regular.

F1
1.2

Its displayed induced P3 has degrees 1,2,1, so it is not regular.

F2
2.1

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

step 1.1step 1.2F3∎

Remarks


\draw[->,line width=.9pt] (2.1,.75)--node[above,font=\scriptsize] {$G[\{v_0,v_1,v_2\}]$} (4.05,.75);

\node[chosen] (w0) at (4.55,0) {$v_0$}; \node[chosen] (w1) at (5.75,1.5) {$v_1$}; \node[chosen] (w2) at (6.95,0) {$v_2$}; \draw[edge] (w0)--(w1)--(w2); \node[caption] at (5.75,-.65) {$P_3$: degrees $1,2,1$}; \end{tikzpicture} ```

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