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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 . Consequently no walk can cross every bridge exactly once.
Facts & Assumptions
Given: The connected four-vertex, seven-edge Konigsberg multigraph with degrees .
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).
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
Counting the bridge ends incident with the four land regions gives degrees , so all four vertices have odd degree.
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.
Depends on
Used by
Nothing in the library uses this result yet.
Dependency tree · two levels
8 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
- Applied Combinatorics, Euler Tours and Trails (standard reference, not scraped)