An Euler circuit with an isolated vertex

Goal and scope

Audit every edge exactly once and the nonisolated support condition.

Finite simple undirected; zero edges use the length-zero-circuit convention.

Givens and symbols

• work

• G1 — V={0,1,2,3}, E={01,12,02};3 is isolated; walk W=(0,1,2,0).

• V,E — complete vertex and edge sets
• W — vertex walk sequence

Rules and sources

Some rule statements have reader clarifications in this revision: they are paraphrases, not verbatim quotations. Referenced original source bytes and hashes are unchanged.

• mko_definition · cited — A trail in a finite simple undirected graph may repeat vertices but not edges. The connectivity part of 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. · mko-eulerian-trail
• work · prerequisite — A finite simple undirected graph has loop-free duplicate-free unordered edges; graph properties are audited on all declared V and E. Each walk step uses an original edge; a trail repeats no edge; an Euler trail uses every edge exactly once; a circuit is a closed trail; degree counts incident edges.

Main derivation

E=\{01,12,02\}

\deg=(2,2,2,0)

\{v:\deg(v)>0\}=\{0,1,2\}

0-1-2-0\text{ connects the entire edge support}

W_0,W_1=(0,1)

\{0,1\}=01\in E

W_1,W_2=(1,2)

\{1,2\}=12\in E

W_2,W_3=(2,0)

\{2,0\}=02\in E

\text{edge sequence }(01,12,02)

\text{distinct edges and edge set exactly }E

W_0=W_3=0

W\text{ is an Euler circuit}

E=\varnothing,\quad V\ne\varnothing

W=(v)\text{ is a length-zero Euler circuit for }v\in V

Substeps and lemmas

No substeps in this guide.

Reverse audits

• S08 · premise_audit · G1, S07, work — The singleton walk repeats no edge and covers empty E.
• S07 · premise_audit · G1, S06, work — Euler concerns all edges, not Hamilton's once-per-vertex requirement.
• S06 · premise_audit · G1, S05, work — Revisiting0 is allowed; a trail forbids repeated edges.
• S05 · premise_audit · G1, S04, work — 02 and 20 denote the same unordered edge.
• S04 · premise_audit · G1, S03, work — It does not reuse01.
• S03 · premise_audit · G1, S02, work — Consecutive walk vertices require an edge.
• S02 · premise_audit · G1, S01, work — The edge-using circuit need not visit3.
• S01 · premise_audit · G1, work — Each undirected edge contributes two to the degree sum.

Error controls and negative controls

Unordered edges

Normalize2-to0 to02 before comparing edges.

Support

Apply connectivity to vertices with positive degree.

• Isolated 3 makes an Euler circuit impossible. → reject — The supplied W already uses every edge.
• A trail cannot revisit0. → reject — A trail forbids repeated edges, not vertices.
• (0,1,0,2) is an Euler trail here. → reject — It uses01 twice and omits12.

Result and limitations

W=(0,1,2,0) uses all three edges once and is an Euler circuit; isolated 3 is harmless.

Verification and source binding

2 declared checks for this guide; only the listed expressions are covered.

• source commit: b1b2f1dcf8c50d67a93fe52d95f8eef7b3abba9f
• adopted source SHA-256: cd1f3d71a3456a1a2c90e91c98b3c616a6c81cfb80bc766effb371a9418f4eed
• original source SHA-256: e58be6b3dd89120f7fd12c07583f1e4197ebffd5d8058f2882ca58fc9b81553c
• Lean: NOT_RUN · independent AI: SCOPED_LEAD_AND_BILINGUAL_CONTEXT_REVIEW

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