False statementConstruction: AI-adaptedVerification: AI-generatedprecheck passaudited 2026-07-31

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FALSE: a finite simple graph is determined up to isomorphism by its degree sequence

Statement

FALSE. A finite simple graph is determined up to isomorphism by its degree sequence.

Facts & Assumptions

Given: The cycle graph $C_{6}$ and the graph $H$ that is the disjoint union of two copies of $C_{3}$ .

[F1]

$C_{n}$ has $n$ vertices arranged in one cycle for $n \geq 3$ (Empty and complete graphs, complete bipartite graphs, and the convention that $P_{n}$ and $C_{n}$ have $n$ vertices).

[F2]

A graph isomorphism is a bijection preserving adjacency in both directions, and therefore sends cycles to cycles of the same length (Graph isomorphisms, automorphisms and graph complements).

[F3]

The degree sequence records the multiset of vertex degrees (Adjacency, incidence, open and closed neighbourhoods, vertex degree, minimum degree and maximum degree).

Refutation

technique · direct

1.1

Every vertex of $C_{6}$ has degree $2$ . Every vertex of $H$ lies on one of its two triangles and also has degree $2$ . Thus both degree sequences are $(2, 2, 2, 2, 2, 2)$ .

givenF1F3

1.2

The graph $H$ contains a cycle of length $3$ , namely either triangle, whereas $C_{6}$ contains no triangle because its only edges join consecutive vertices on its six-cycle.

givenF1

2.1

If $C_{6} ≅ H$ , [F2] would carry a triangle of $H$ to a triangle of $C_{6}$ , contradicting step 1.2. Hence the graphs have the same degree sequence but are not isomorphic, refuting the claim.

step 1.1step 1.2F2∎

Depends on

Used by

C₆ and the disjoint union of two triangles have the same degree sequence but are not isomorphic Counterexample

Dependency tree · two levels

9 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)