CounterexampleConstruction: AI-adaptedVerification: AI-generatedprecheck passaudited 2026-07-31

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Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
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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

1.1

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 $G ≇ H$ . They have the same degree sequence by step 1.1, which refutes the stated claim.

step 1.1step 1.2F2∎

Remarks


% C_3 dot-union C_3. \node[vertex] (h1) at (3.3,1) {}; \node[vertex] (h2) at (4.35,-.8) {}; \node[vertex] (h3) at (2.25,-.8) {}; \node[vertex] (j1) at (6.2,1) {}; \node[vertex] (j2) at (7.25,-.8) {}; \node[vertex] (j3) at (5.15,-.8) {}; \draw[edge] (h1)--(h2)--(h3)--cycle; \draw[edge] (j1)--(j2)--(j3)--cycle; \node[caption] at (4.75,-2) {$H=C_3\mathbin{\dot\cup}C_3$\\two components}; \node[font=\scriptsize] at (2.35,1.85) {every displayed vertex has degree $2$}; \end{tikzpicture} ```

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