# Euler trails and circuits

This English edition is an AI candidate and does not yet have an OCME Atlas (GPT-6) Twin closure receipt.

An Euler trail uses each edge once; on connected nonzero-edge support, zero odd-degree vertices give a circuit and two give an open Euler trail.

mko-eulerian-trail · definition · v0.10.0



## Mathematical statement

A trail in a finite simple undirected graph may repeat vertices but not edges. The existence criterion requires only that all nonisolated vertices lie in one component; isolated vertices do not break the edge support. A nonempty vertex set with no edges uses the length-zero-circuit convention; the empty-vertex graph is out of scope.





## Representative formula (exact TeX)

deg(v)





## Assumptions


- The complete finite undirected loop-free simple graph and its edge list are explicit.
- Fix the zero-edge convention as a length-zero circuit on a nonempty vertex carrier.





## Symbol scope


- \deg — vertex degree
- T — Euler trail





## Intuition, definition, and misconceptions

A trail prohibits repeated edges, not revisited vertices.

Under the support-connectivity premise, zero odd vertices give a circuit and two give an open trail.

The criterion acts on nonisolated support and supplies no Hamiltonian-path or cycle claim.

Do not treat a trail as a simple path or let isolated vertices alone invalidate an Euler trail.





## Adopted derivation guide

[An Euler circuit with an isolated vertex](https://ocme.evemisslab.com/en/derivations/derivation-mko-eulerian-trail-01/)The original MKO is retained. The guide has its own append-only revision and verification boundaries, not a new Lean proof.





## Prerequisites


- [Graph walks and simple paths](https://ocme.evemisslab.com/en/mko/mko-graph-walk-simple-path/) — A trail and a simple path have different repetition rules.
- [Finite simple undirected graphs](https://ocme.evemisslab.com/en/mko/mko-graph/) — All vertices and edges are supplied.
- [Connected component](https://ocme.evemisslab.com/en/mko/mko-connected-component/) — Connectivity is checked on nonisolated support.





## Verification boundary

Only finite simple undirected graphs are handled; directed, multigraph and Hamiltonian problems are separate.

not_formalized

This does not mean the content is false; it means no replayable computational or formal evidence is registered yet.





## Next obligation

Add Hierholzer construction and directed Euler criteria.





## Same-source structured data

This page and the interactive view use the same object and existing language edition; this adds no proof or human review.

[Canonical MKO JSON](https://ocme.evemisslab.com/data/mko/eulerian-trail.json) · [Plain Markdown](https://ocme.evemisslab.com/en/mko/mko-eulerian-trail/index.md)