LemmaStatement: AI-adaptedProof: AI-generatedSession-authored (Fable 5 assisted)precheck passaudited 2026-08-02

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The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
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In an even multigraph or balanced digraph, a trail maximal among unused incident edges returns to its start

Statement

Let $T$ be a nonempty trail in a finite undirected multigraph in which every vertex has even degree. If $T$ cannot be extended at its terminal vertex by an unused incident edge, then $T$ is closed.

Likewise, let $T$ be a nonempty directed trail in a finite digraph satisfying $\deg^+(v)=\deg^-(v)$ at every vertex. If $T$ cannot be extended at its terminal vertex by an unused outgoing arc, then $T$ is closed.

Facts & Assumptions

Given: One of the two finite trails in the statement, with initial vertex $s$ and terminal vertex $t$ .

[F1]

Trails do not repeat edges or arcs, and they are closed exactly when $s=t$ (Euler trails and Euler circuits in multigraphs and digraphs).

[F2]

An undirected loop contributes two to degree; a directed loop contributes one to both indegree and outdegree (Degree in a multigraph, indegree and outdegree in a digraph, and their underlying connectivity).

Proof

technique · direct

1.1

In the undirected case, if $s\ne t$ , the used edge ends at $t$ occur in pairs for every intermediate visit and with one additional final arrival. Thus an odd number of edge ends incident with $t$ have been used; loops contribute pairs by [F2]. Since $\deg(t)$ is even, at least one incident edge is unused, contradicting maximality.

givenF1F2algebra

1.2

In the directed case, if $s\ne t$ , the trail uses one more incoming than outgoing arc at $t$ . Total indegree equals total outdegree, so the number of unused outgoing arcs at $t$ is one more than the number of unused incoming arcs and is positive, again contradicting maximality.

givenF1F2algebra

2.1

Therefore $s=t$ in either case, so the trail is closed.

step 1.1step 1.2F1cases∎

Depends on

Used by

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 19 results over 14 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)