ExampleConstruction: AI-generatedVerification: AI-generatedprecheck passaudited 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.

$K_{3, 3}$ has a Hamilton cycle

Example

Let the two parts of $K_{3, 3}$ be $A = {a_{1}, a_{2}, a_{3}}$ and $B = {b_{1}, b_{2}, b_{3}}$ . Then

$a_{1}, b_{1}, a_{2}, b_{2}, a_{3}, b_{3}, a_{1}$

is a Hamilton cycle.

Facts & Assumptions

Given: The displayed bipartition of $K_{3, 3}$ .

[F1]

Every vertex in $A$ is adjacent to every vertex in $B$ , and there are no edges within a part (Empty and complete graphs, complete bipartite graphs, and the convention that $P_{n}$ and $C_{n}$ have $n$ vertices).

[F2]

A Hamilton cycle visits every vertex exactly once before returning to its initial vertex (Hamilton paths, Hamilton cycles, Hamiltonian graphs and Hamilton-connected graphs).

Verification

technique · direct

1.1

Each consecutive pair in the displayed cyclic list has one endpoint in $A$ and one in $B$ , so all six required edges occur by [F1].

2.1

Before returning to $a_{1}$ , the list contains each of the six vertices exactly once. It is therefore a Hamilton cycle by [F2].

step 1.1F2∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

7 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

Applied Combinatorics, Eulerian and Hamiltonian Graphs (standard reference, not scraped)

K3,3K3,3​ has a Hamilton cycle