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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
Statement
A connected finite undirected multigraph has an Euler trail if and only if the number of odd-degree vertices is zero or two. It has an open Euler trail, one whose endpoints are distinct, if and only if it has exactly two odd-degree vertices; those vertices are the endpoints.
Facts & Assumptions
Given: A connected finite undirected multigraph .
An Euler trail uses every edge exactly once and is an Euler circuit when closed (Euler trails and Euler circuits in multigraphs and digraphs).
Loops count twice toward undirected degree (Degree in a multigraph, indegree and outdegree in a digraph, and their underlying connectivity).
A connected finite undirected multigraph has an Euler circuit exactly when all degrees are even (Euler's theorem and Hierholzer's construction: a connected finite undirected multigraph has an Euler circuit if and only if every degree is even).
Proof
Along any trail, every vertex other than two distinct endpoints uses incident edge ends in pairs. If the trail is open, its two endpoints each use one unpaired end; if it is closed, every used incidence is paired. Loops contribute pairs by [F2].
If every degree is even, [L1] supplies an Euler circuit, which is an Euler trail.
If exactly and have odd degree, add one distinguished edge between and , allowing it to be parallel to an existing edge. The augmented connected multigraph has all degrees even, so [L1] supplies an Euler circuit. Rotate that circuit to begin with the distinguished edge and delete it; what remains is an Euler trail in from to .
Therefore an Euler trail forces exactly two odd-degree vertices when open and zero odd-degree vertices when closed.
Steps 2.1, 1.2, and 1.3 prove the criterion and show that openness occurs exactly in the two-odd-vertex case.
Depends on
Used by
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
- Applied Combinatorics, Euler Tours and Trails (standard reference, not scraped)