CounterexampleConstruction: AI-adaptedVerification: AI-generatedSession-authored (Fable 5 assisted)precheck passaudited 2026-07-31

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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 $H=C_3\mathbin{\dot\cup}C_3$ , 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

1.1

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 $(2,2,2,2,2,2)$ .

givenF1

1.2

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.

givenF1

2.1

By [F2], connectedness is invariant under isomorphism, so $G\not\cong H$ . They have the same degree sequence by step 1.1, which refutes the stated claim.

step 1.1step 1.2F2∎

Remarks

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 15 results over 8 levels. An arrow runs from a result to what uses it, and this result sits at the bottom with a heavier outline. Click the chart to enlarge it.

Sources

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

C6C_6C6​ and the disjoint union of two triangles have the same degree sequence but are not isomorphic