# Question 15530: Isolated Vertices Still Belong to Exactly One Part [Return to catalogue page 78](https://ocme.evemisslab.com/en/questions/page/78/) [← Question derivation index](https://ocme.evemisslab.com/en/questions/) · [Original question batch](https://ocme.evemisslab.com/en/practice/batch-20261014-advance-20261031-r02-bipartite-graph-foundations/#q-20261014-advance-20261031-r02-bipartite-graph-0080) This teaching derivation has second-context AI mathematical and bilingual semantic review, limited to the model and assumptions stated in this question. It has no human review or new Lean proof and does not promote the original question or MKO to Canonical truth. R0030-15530 · q-20261014-advance-20261031-r02-bipartite-graph-0080 ## Original question and givens Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with edges=[[0, 1], [1, 2], [2, 3], [3, 4]]. Normalize each edge increasingly, then sort edges lexicographically; reversed duplicates and loops are forbidden. Keep isolated vertices in the carrier. No directed-graph conclusion is inferred. The total part assignment=[0, 0, 1, 0, 0, 0] uses labels 0 or 1 for every vertex. Its two inverse-image parts are disjoint and cover the whole carrier. Every edge must cross parts; empty parts are allowed in an edgeless graph. Failure of this assignment alone does not prove the graph nonbipartite. vertex=5 must be isolated. Give (vertex,incident_edges,assigned_part,whole_partition_proper?,isolated_vertex_may_be_in_either_part_and_empty_part_allowed_for_edgeless_graph). - G001 — Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with edges=[[0, 1], [1, 2], [2, 3], [3, 4]]. Normalize each edge increasingly, then sort edges lexicographically; reversed duplicates and loops are forbidden. Keep isolated vertices in the carrier. No directed-graph conclusion is inferred. The total part assignment=[0, 0, 1, 0, 0, 0] uses labels 0 or 1 for every vertex. Its two inverse-image parts are disjoint and cover the whole carrier. Every edge must cross parts; empty parts are allowed in an edgeless graph. Failure of this assignment alone does not prove the graph nonbipartite. vertex=5 must be isolated. Give (vertex,incident_edges,assigned_part,whole_partition_proper?,isolated_vertex_may_be_in_either_part_and_empty_part_allowed_for_edgeless_graph). ## Definition and scope Use the original question domain, assumptions and cited MKO definitions. This item has second-context mathematical and bilingual review, with the model, answer and every arithmetic step independently recomputed from the original parameters; the source question remains a candidate. - [mko-bipartite-graph](https://ocme.evemisslab.com/en/mko/mko-bipartite-graph/) ## Step-by-step derivation S001 · impliesBefore complete undirected vertex carrier Fin(6);original edges=[[0, 1], [1, 2], [2, 3], [3, 4]] After normalized sorted simple edges=[[0, 1], [1, 2], [2, 3], [3, 4]];all vertices including isolates=[0, 1, 2, 3, 4, 5] Conditions: none Normalize endpoints and then sort edges lexicographically.Reversed duplicates are still duplicates and are rejected,as are loops.The complete carrier includes vertices absent from the edge list. Step identifiers and dependenciesRule: graph_complete_simple_carrier_normalized_edge_order · Uses: G001 S002 · impliesBefore total original part labels=[0, 0, 1, 0, 0, 0];complete carrier Fin(6) After part0=[0, 1, 3, 4, 5];part1=[2];disjoint full cover=Yes Conditions: none The two inverse images of0 and1 are disjoint and cover every originally labeled vertex.An empty part is allowed for an edgeless graph;isolates remain included.Edge crossing is still a separate obligation. Step identifiers and dependenciesRule: bipartition_total_inverse_image_parts · Uses: G001, S001 S003 · impliesBefore parts:normalized edge index=0;edge=[0, 1];endpoint labels=[0, 0] After different endpoints=No;same-label conflict=Yes Conditions: none Check the endpoints of each actual edge.Reused labels at nonadjacent vertices are not conflicts,and one crossing edge does not certify all other edges. Step identifiers and dependenciesRule: graph_every_edge_endpoint_constraint · Uses: G001, S002 S004 · impliesBefore parts:normalized edge index=1;edge=[1, 2];endpoint labels=[0, 1] After different endpoints=Yes;same-label conflict=No Conditions: none Check the endpoints of each actual edge.Reused labels at nonadjacent vertices are not conflicts,and one crossing edge does not certify all other edges. Step identifiers and dependenciesRule: graph_every_edge_endpoint_constraint · Uses: G001, S003 S005 · impliesBefore parts:normalized edge index=2;edge=[2, 3];endpoint labels=[1, 0] After different endpoints=Yes;same-label conflict=No Conditions: none Check the endpoints of each actual edge.Reused labels at nonadjacent vertices are not conflicts,and one crossing edge does not certify all other edges. Step identifiers and dependenciesRule: graph_every_edge_endpoint_constraint · Uses: G001, S004 S006 · impliesBefore parts:normalized edge index=3;edge=[3, 4];endpoint labels=[0, 0] After different endpoints=No;same-label conflict=Yes Conditions: none Check the endpoints of each actual edge.Reused labels at nonadjacent vertices are not conflicts,and one crossing edge does not certify all other edges. Step identifiers and dependenciesRule: graph_every_edge_endpoint_constraint · Uses: G001, S005 S007 · impliesBefore original vertex=5;all complete graph edges=[[0, 1], [1, 2], [2, 3], [3, 4]];assigned part=0 After all incident edges=[];whole partition proper=No;isolate stays in exactly one part Conditions: none Only after checking no incident edges may this isolate be put in either part.It remains in the complete carrier with exactly one label.Changing it neither affects edge constraints nor repairs unrelated conflicts. Step identifiers and dependenciesRule: bipartite_isolated_vertex_not_omitted_or_whole_graph_fixed · Uses: G001, S006 ## Reverse checks - S007 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S006 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S005 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S004 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S003 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S002 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S001 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. ## Result (5, [], 0, No, isolated_vertex_may_be_in_either_part_and_empty_part_allowed_for_edgeless_graph) The full substitutions and premise checks give the original answer; source identity,format and status are unchanged. ## This question review and source binding Review date: 2026-10-26 Original questions, answers, step expressions, numeric ASTs and historical records are retained. Source candidate labels remain historical and are not overwritten as formal proof. Source hashes and review receipts - input commit: 0f7be61e2c22317a4075561dc822c87d881a50db - original question SHA-256: 2c26b94a0b8df91f9af8a9ba74229fea38c0f3de10ee5b830f5b716fb23f3214 - candidate record SHA-256: 2dd85e12bfbb8c5149c16904f1d223832720de71739ed09368e400409edcf852 - semantic review receipt SHA-256: 73bc08066ffb1a81eacb2d4e288d03f522aeefd3949fa0d97da41b484ed737bc - question-first witness SHA-256: ac120452a208b257b8270f38351a09dcd832cdbc8d9db6b69933f83e43cc9336 - Lean: NOT_RUN · human review: NOT_PERFORMED [Same-source full JSON](https://ocme.evemisslab.com/data/question-derivations/questions/q-20261014-advance-20261031-r02-bipartite-graph-0080.json)