False statementConstruction: AI-adaptedVerification: AI-generatedSession-authored (Fable 5 assisted)precheck passverified 2026-08-04 (gpt-5.6-sol-codex-subscription)

How statement and proof provenance work

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.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

FALSE: a connected simple graph in which every vertex has even degree must itself be a cycle

Statement

FALSE. Every connected finite simple graph in which every vertex has even degree is a cycle graph.

Facts & Assumptions

Given: The bowtie graph with vertices $x,a,b,c,d$ and edges $xa,ab,bx,xc,cd,dx$ , consisting of two triangles that share only $x$ .

[F1]

Degree counts the neighbours of a vertex (Adjacency, incidence, open and closed neighbourhoods, vertex degree, minimum degree and maximum degree).

[F2]

A connected graph has a path between every two vertices (Connected graphs and connected components defined by the existence of vertex paths).

[F3]

In the cycle graph $C_n$ each vertex is joined by an edge exactly to its predecessor and its successor modulo $n$ (Empty and complete graphs, complete bipartite graphs, and the convention that $P_n$ and $C_n$ have $n$ vertices); with [F1] this gives every vertex of a cycle graph exactly two neighbours.

Refutation

technique · direct

1.1

Every two vertices of the bowtie are joined by a path, using $x$ when they lie in different triangles, so the graph is connected by [F2].

givenF2

1.2

The vertex $x$ has neighbours $a,b,c,d$ and hence degree $4$ ; each other vertex has degree $2$ . Thus all degrees are even by [F1].

givenF1

2.1

The bowtie is not a cycle graph because its vertex $x$ has degree $4$ , whereas [F3] gives degree $2$ at every vertex of a cycle graph. Steps 1.1 and 1.2 therefore refute the claim.

step 1.1step 1.2F3∎

Depends on

Used by

Two triangles sharing one vertex form a connected simple graph with all degrees even that is not a cycle Counterexample

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 17 results over 11 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

Cornell Mathematics, Graph Theory notes (standard reference, not scraped)