CounterexampleConstruction: AI-adaptedVerification: AI-generatedprecheck pass

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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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$C_{6}$ and the disjoint union of two triangles have the same degree sequence but are not isomorphic

Statement refuted

The false statement FALSE: a finite simple graph is determined up to isomorphism by its degree sequence claims that a finite simple graph is determined up to isomorphism by its degree sequence.

Facts & Assumptions

Given: $G = C_{6}$ and $H = C_{3} \dot{\cup} C_{3}$ , the disjoint union of two triangles.

[F1]

$C_{n}$ is the cycle graph on $n$ vertices (Empty and complete graphs, complete bipartite graphs, and the convention that $P_{n}$ and $C_{n}$ have $n$ vertices).

[F2]

Isomorphisms preserve adjacency and therefore preserve path-reachability and connectedness (Graph isomorphisms, automorphisms and graph complements, Connected graphs and connected components defined by the existence of vertex paths).

Counterexample

technique · direct

1.1givenF1

Every vertex of $G$ lies on the six-cycle and has degree $2$ . Every vertex of $H$ lies on one of its two triangles and has degree $2$ . Thus both degree sequences are $(2, 2, 2, 2, 2, 2)$ .

1.2givenF1

The graph $G$ is connected, since either direction around the cycle gives a path between any two vertices. The graph $H$ is disconnected, since no edge joins its two triangles.

2.1step 1.1step 1.2F2∎

By [F2], connectedness is invariant under isomorphism, so $G ≇ H$ . They have the same degree sequence by step 1.1, which refutes the stated claim.

Remarks

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

12 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

R. Diestel, Graph Theory, Chapter 1 preview (standard reference, not scraped)

C6C6​ and the disjoint union of two triangles have the same degree sequence but are not isomorphic