Euler trails and circuits

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

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 — A trail and a simple path have different repetition rules.
• Finite simple undirected graphs — All vertices and edges are supplied.
• 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.