Use the complete bounded simple graph on Fin(6)={0,…,5} with canonical ordered undirected edges=[[0, 1], [2, 3]]. Edge indices are zero-based positions in this exact list. A matching contains distinct edge indices with no shared endpoint. Enumerate every edge subset to determine the global maximum size; include the empty matching. List indices within each matching increasingly and enumerate matchings in increasing edge-subset mask order. Maximum,inclusion-maximal,perfect and unique are distinct properties. Give (matching_count,maximum_size,maximum_matchings,exhaustive_all_edge_subsets).
G001 — Use the complete bounded simple graph on Fin(6)={0,…,5} with canonical ordered undirected edges=[[0, 1], [2, 3]]. Edge indices are zero-based positions in this exact list. A matching contains distinct edge indices with no shared endpoint. Enumerate every edge subset to determine the global maximum size; include the empty matching. List indices within each matching increasingly and enumerate matchings in increasing edge-subset mask order. Maximum,inclusion-maximal,perfect and unique are distinct properties. Give (matching_count,maximum_size,maximum_matchings,exhaustive_all_edge_subsets).
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.
complete original carrier Fin(6);ALREADY canonical original edge list=[[0, 1], [2, 3]]
After
loop-free unique original edges=2<=10;all edge subsets have mask0 through3
Conditions: none
This family's bounds are n≤12 and at most10 edges,permitting exhaustive enumeration.Do not reuse the earlier n30 guard.Input endpoints and edges must already be canonical,with indices bound to original edges and no loops or duplicates.
original edge-subset integer mask=0;increasing original indices=[];actual selected edges=[]
After
used endpoint union=[];repeated endpoints in traversal order=[];valid matching=Yes;size=0
Conditions: none
Enumerate integer masks in order,including the empty subset,and check endpoint conflicts first.Invalid collections cannot enter the optimum comparison.Every original edge subset has one mask,including alternatives that replace selected edges.
original edge-subset integer mask=1;increasing original indices=[0];actual selected edges=[[0, 1]]
After
used endpoint union=[0, 1];repeated endpoints in traversal order=[];valid matching=Yes;size=1
Conditions: none
Enumerate integer masks in order,including the empty subset,and check endpoint conflicts first.Invalid collections cannot enter the optimum comparison.Every original edge subset has one mask,including alternatives that replace selected edges.
original edge-subset integer mask=2;increasing original indices=[1];actual selected edges=[[2, 3]]
After
used endpoint union=[2, 3];repeated endpoints in traversal order=[];valid matching=Yes;size=1
Conditions: none
Enumerate integer masks in order,including the empty subset,and check endpoint conflicts first.Invalid collections cannot enter the optimum comparison.Every original edge subset has one mask,including alternatives that replace selected edges.
original edge-subset integer mask=3;increasing original indices=[0, 1];actual selected edges=[[0, 1], [2, 3]]
After
used endpoint union=[0, 1, 2, 3];repeated endpoints in traversal order=[];valid matching=Yes;size=2
Conditions: none
Enumerate integer masks in order,including the empty subset,and check endpoint conflicts first.Invalid collections cannot enter the optimum comparison.Every original edge subset has one mask,including alternatives that replace selected edges.
global maximum size=2;ALL maxima in ORIGINAL MASK order=[[0, 1]]
Conditions: none
Every valid candidate has size at most this maximum,and every attaining witness is retained.This supplies both a global upper bound and attainment.Keep mask order rather than lexicographic list order,and infer no general complexity theorem.
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 — Recompute exhaustive matching validity indicator count with exact rational arithmetic, not rounded decimals.
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
(4, 2, [[0, 1]], exhaustive_all_edge_subsets)
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.