CounterexampleConstruction: AI-generatedVerification: AI-generatedprecheck passaudited 2026-08-02

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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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$P_{3}$ has a Hamilton path but no Hamilton cycle

Statement refuted

Every finite simple graph with a Hamilton path has a Hamilton cycle.

Counterexample

The path graph $P_{3}$ has vertices $0, 1, 2$ and edges $01, 12$ .

Facts & Assumptions

Given: The path graph $P_{3}$ .

[F1]

The graph $P_{3}$ has the displayed vertices and consecutive edges (Empty and complete graphs, complete bipartite graphs, and the convention that $P_{n}$ and $C_{n}$ have $n$ vertices).

[F2]

A Hamilton path contains every vertex exactly once (Hamilton paths, Hamilton cycles, Hamiltonian graphs and Hamilton-connected graphs).

[L1]

A Hamiltonian graph satisfies the vertex-deletion component bound (If $G$ has a Hamilton cycle, then deleting any nonempty proper vertex set $S$ leaves at most $∣ S ∣$ connected components).

Verification

technique · direct

1.1

The path $0, 1, 2$ uses both edges and contains every vertex exactly once, so it is a Hamilton path.

F1F2

1.2

Deleting the middle vertex $1$ leaves two isolated vertices, hence two components. This violates [L1] for the singleton set $S = {1}$ , so $P_{3}$ has no Hamilton cycle.

F1L1algebra

2.1

Therefore a Hamilton path need not extend to a Hamilton cycle.

step 1.1step 1.2∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

10 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)

P3P3​ has a Hamilton path but no Hamilton cycle