CounterexampleConstruction: AI-adaptedVerification: AI-generatedprecheck 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 $x a, a b, b x, x c, c d, d x$ .

[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 $x a b x$ and $x c d x$ . 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

tikz \begin{tikzpicture}[ vertex/.style={draw,circle,fill=white,minimum size=8mm,inner sep=1pt,font=\small}, centre/.style={vertex,draw=red!75!black,fill=red!12,line width=1.4pt}, edge/.style={draw=gray!75,line width=1.1pt} ] \node[centre] (x) at (0,0) {$x$}; \node[vertex] (a) at (-2,1.25) {$a$}; \node[vertex] (b) at (-2,-1.25) {$b$}; \node[vertex] (c) at (2,1.25) {$c$}; \node[vertex] (d) at (2,-1.25) {$d$}; \draw[edge] (x)--(a)--(b)--(x); \draw[edge] (x)--(c)--(d)--(x); \node[font=\scriptsize,red!75!black,anchor=west] at (.45,.2) {$\deg(x)=4$}; \node[font=\scriptsize] at (0,-1.9) {$\deg(a)=\deg(b)=\deg(c)=\deg(d)=2$}; \end{tikzpicture}

Depends on

Used by

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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

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