A bipartition certificate with an isolated vertex

Goal and scope

Audit every bipartition obligation and distinguish one failed partition from nonbipartiteness.

Finite simple undirected graphs; empty parts are allowed but coverage remains required.

Givens and symbols

• work

• G1 — V={0,1,2,3}, E={{0,1},{1,2}}; candidate A={0,2,3}, B={1}.

• V,E — complete vertex and edge sets
• A,B — the two parts

Rules and sources

• mko_definition · cited — Check complete vertex coverage, disjointness of the two parts and every edge crossing. A failed proposed partition does not prove nonbipartiteness; an odd cycle can certify it. Isolated vertices may belong to either part, and whether empty parts are allowed must be explicit. · mko-bipartite-graph
• work · prerequisite — A finite simple undirected graph has loop-free duplicate-free unordered edges; graph properties are audited on all declared V and E. A bipartition certificate has disjoint parts with union V and every edge crossing.

Main derivation

V,E

E\subseteq\{\{u,v\}:u,v\in V,\ u\ne v\}

A=\{0,2,3\},\quad B=\{1\}

A\cap B=\varnothing

A\cup B=\{0,1,2,3\}

A\cup B=V

\{0,1\}\in E

0\in A,\quad1\in B

\{1,2\}\in E

1\in B,\quad2\in A

A'=\{0,1\},\quad B'=\{2,3\}

\{0,1\}\subseteq A'\implies(A',B')\text{ invalid}

H:0-1-2-0\quad\text{(separate triangle)}

c(0)\ne c(1)\ne c(2)\implies c(2)=c(0),\ c(2)\ne c(0)

A\cap B=\varnothing,\quad A\cup B=V,\quad\text{all edges cross}

G\text{ is bipartite}

Substeps and lemmas

No substeps in this guide.

Reverse audits

• S08 · premise_audit · G1, S07, mko_definition — An edgeless graph may use(V,empty), but may not omit vertices.
• S07 · premise_audit · G1, S06, work — This odd-cycle refutation concerns the separate complete triangle, not the original graph.
• S06 · premise_audit · G1, S05, work — One invalid candidate cannot refute existence of a valid one.
• S05 · premise_audit · G1, S04, work — 0 and 2 may share a part because they are not adjacent.
• S04 · premise_audit · G1, S03, work — Vertex counts alone do not prove this edge obligation.
• S03 · premise_audit · G1, S02, work — Omitting an isolate would still invalidate the partition.
• S02 · premise_audit · G1, S01, work — A vertex cannot belong to both parts.
• S01 · premise_audit · G1, work — These are the only edges; no0-to 2 edge is implicit.

Error controls and negative controls

All vertices

Assign isolated 3 to a part.

Existential quantifier

A valid witness proves existence; one failure does not prove nonexistence.

• Isolated 3 may be omitted. → reject — The union would no longer equal V.
• Failure of A' proves nonbipartiteness. → reject — A,B already form a valid partition.
• A triangle admits two parts. → reject — Alternation around three edges forces adjacent endpoints into the same part.

Result and limitations

A={0,2,3}, B={1} certify the graph bipartite.

Verification and source binding

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

• source commit: b1b2f1dcf8c50d67a93fe52d95f8eef7b3abba9f
• adopted source SHA-256: bd3b9a767549e7cefb60cebc2278fb332f5bac033b84321bd0c2eb7004197c3c
• original source SHA-256: b94470821c9e899e02832a1b63f4ceaf9869bac586a0967c66175e03ad73bf37
• Lean: NOT_RUN · independent AI: SCOPED_LEAD_AND_BILINGUAL_CONTEXT_REVIEW

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