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ExampleConstruction: AI-adaptedVerification: AI-generatedSession-authored (Fable 5 assisted)precheck passaudited 2026-08-02
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The Konigsberg bridges multigraph has four odd-degree vertices and no Euler trail

Example

Model the four land regions in the Konigsberg bridges problem by four vertices, with one multigraph edge for each of the seven bridges. The four vertex degrees are 5,3,3,35,3,3,3. Consequently no walk can cross every bridge exactly once.

ABCD3353

Facts & Assumptions

Given: The connected four-vertex, seven-edge Konigsberg multigraph with degrees 5,3,3,35,3,3,3.

[F1]

Parallel bridges are distinct multigraph edges, and degree counts their incident ends (Degree in a multigraph, indegree and outdegree in a digraph, and their underlying connectivity).

[L1]

A connected finite multigraph has an Euler trail exactly when it has zero or two odd-degree vertices (A connected finite undirected multigraph has an Euler trail if and only if it has zero or two odd-degree vertices; an open Euler trail occurs exactly in the two-vertex case).

Verification

technique · direct
1.1

Counting the bridge ends incident with the four land regions gives degrees 5,3,3,35,3,3,3, so all four vertices have odd degree.

givenF1algebra
2.1

Since four is neither zero nor two, [L1] rules out an Euler trail. Such a trail would be exactly a crossing of every bridge once.

step 1.1L1∎

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: 22 results over 16 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