LemmaStatement: AI-adaptedProof: AI-generatedprecheck passaudited 2026-08-02

How statement and proof provenance work

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.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

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 $s \neq t$ , 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 $s \neq t$ , 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 · 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)