False statementConstruction: AI-adaptedVerification: AI-generatedprecheck passverified 2026-08-04 (gpt-5.6-sol-codex-subscription)

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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 $x a, a b, b x, x c, c d, d x$ , 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 · two levels

10 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

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