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CounterexampleConstruction: AI-generatedVerification: AI-generatedprecheck passaudited 2026-07-31
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Two dense lobes meeting at one cut vertex give κ(G)=1<λ(G)=2<δ(G)=3

Statement refuted

The false statement FALSE: vertex connectivity, edge connectivity and minimum degree are always equal claims that vertex connectivity, edge connectivity and minimum degree always agree.

Facts & Assumptions

Given: For i=1,2, take vertices ai,bi,ci,di spanning K4 with the edge aibi deleted. Add one vertex v, add the edges vai,vbi for both i, and add no edge between the two four-vertex lobes.

[F1]

κ and λ are the least sizes of vertex and edge cuts (Vertex cuts, edge cuts, vertex connectivity κ(G) and edge connectivity λ(G), with conventions for complete and one-vertex graphs).

[F2]

δ(G) is the least vertex degree (Adjacency, incidence, open and closed neighbourhoods, vertex degree, minimum degree and maximum degree).

Counterexample

technique · direct
1.1

The graph is connected, and deleting v separates the two lobes. No deletion of zero vertices disconnects a connected graph, so κ(G)=1.

givenF1
1.2

In each lobe, ai and bi have two neighbours inside the lobe and the neighbour v, so degree 3; ci,di have degree 3 inside the lobe; and v has degree 4. Hence δ(G)=3.

givenF2
1.3

Deleting va1 and vb1 separates the first lobe from the rest, so λ(G)≤2.

givenF1
1.4

Every edge lies on a cycle. The edges vai and vbi lie on the 4-cycle v,ai,ci,bi,v. Each internal edge incident with ai lies on the triangle ai,ci,di,ai, and each internal edge incident with bi lies on bi,ci,di,bi. Thus deleting one edge leaves an alternate path between its endpoints and cannot disconnect the graph, so λ(G)≥2.

givenF1
2.1

Steps 1.3 and 1.4 give λ(G)=2. Together with steps 1.1 and 1.2, this proves κ(G)=1<λ(G)=2<δ(G)=3 and refutes equality in both Whitney inequalities.

step 1.1step 1.2step 1.3step 1.4∎

Remarks

tikz \begin{tikzpicture}[ vertex/.style={draw,circle,fill=white,minimum size=7.5mm,inner sep=1pt,font=\scriptsize}, cutvertex/.style={vertex,draw=violet!80!black,fill=violet!15,line width=1.4pt}, edge/.style={draw=gray!70,line width=.9pt}, edgecut/.style={draw=orange!85!black,line width=1.7pt} ] \node[cutvertex] (v) at (0,0) {$v$}; % First K_4-a_1b_1 lobe. \node[vertex] (a1) at (-1.5,.75) {$a_1$}; \node[vertex] (b1) at (-1.5,-.75) {$b_1$}; \node[vertex] (c1) at (-3.2,1.05) {$c_1$}; \node[vertex] (d1) at (-3.2,-1.05) {$d_1$}; \draw[edge] (a1)--(c1) (a1)--(d1) (b1)--(c1) (b1)--(d1) (c1)--(d1); \draw[edgecut] (v)--(a1) (v)--(b1); % Second K_4-a_2b_2 lobe. \node[vertex] (a2) at (1.5,.75) {$a_2$}; \node[vertex] (b2) at (1.5,-.75) {$b_2$}; \node[vertex] (c2) at (3.2,1.05) {$c_2$}; \node[vertex] (d2) at (3.2,-1.05) {$d_2$}; \draw[edge] (a2)--(c2) (a2)--(d2) (b2)--(c2) (b2)--(d2) (c2)--(d2); \draw[edge] (v)--(a2) (v)--(b2); \node[font=\scriptsize,violet!80!black] at (0,1.05) {vertex cut $\{v\}$}; \node[font=\scriptsize,orange!85!black] at (-1.25,-1.65) {edge cut $\{va_1,vb_1\}$}; \node[font=\scriptsize] at (0,-2.15) {$\kappa(G)=1<\lambda(G)=2<\delta(G)=3$}; \end{tikzpicture}

Depends on

Used by

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Dependency tree · two levels

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Sources