False statementConstruction: AI-adaptedVerification: AI-generatedSession-authored (Fable 5 assisted)precheck 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 $n\ge3$ (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 $C_6\cong H$ , [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 · next 3 levels

Direct dependencies and their dependencies through the next three levels: 17 results over 10 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)