Original question and givens
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 (maximum_size,maximum_matchings,count,multiple?), listing every maximum in the declared order.
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 (maximum_size,maximum_matchings,count,multiple?), listing every maximum in the declared order.
Step-by-step derivation
S001 · implies
Before
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.
Step identifiers and dependencies
Rule: maxmatching_complete_bounded_original_graph_mask_domain · Uses: G001
S002 · implies
Before
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.
Step identifiers and dependencies
Rule: maxmatching_every_mask_validity_before_global_comparison · Uses: G001, S001
S003 · implies
Before
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.
Step identifiers and dependencies
Rule: maxmatching_every_mask_validity_before_global_comparison · Uses: G001, S002
S004 · implies
Before
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.
Step identifiers and dependencies
Rule: maxmatching_every_mask_validity_before_global_comparison · Uses: G001, S003
S005 · implies
Before
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.
Step identifiers and dependencies
Rule: maxmatching_every_mask_validity_before_global_comparison · Uses: G001, S004
S006 · calculation
Before
exhaustive matching validity indicator count := sum(1, 1, 1, 1)
After
exhaustive matching validity indicator count = 4
Conditions: none
Compute exactly from original roles and established intermediates,retaining denominators,signs,coordinates and indices.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S005
S007 · implies
Before
all4 original masks examined;valid matching count=4;all valid sizes=[0, 1, 1, 2]
After
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.
Step identifiers and dependencies
Rule: maxmatching_exhaustive_upper_bound_and_attaining_all_witnesses · Uses: G001, S006