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

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

Statement refuted

The false statement FALSE: a connected simple graph in which every vertex has even degree must itself be a cycle claims that every connected simple graph whose degrees are all even is itself a cycle.

Facts & Assumptions

Given: The graph $G$ with vertices $x,a,b,c,d$ and edges $xa,ab,bx,xc,cd,dx$ .

[F1]

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

[F2]

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 degree $2$ .

Counterexample

technique · direct

1.1

The edges form the triangles $xabx$ and $xcdx$ . Every vertex reaches $x$ within its triangle, so every two vertices are joined by a path and $G$ is connected.

given

1.2

The vertex $x$ has degree $4$ , while $a,b,c,d$ each have degree $2$ . Thus every degree is even by [F1].

givenF1

2.1

Since $x$ has degree $4$ , [F2] shows that $G$ is not a cycle graph. Steps 1.1 and 1.2 supply the required connected even-degree counterexample.

step 1.1step 1.2F2∎

Remarks

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 18 results over 12 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)