Original question and givens
Use the complete finite loop-free simple undirected graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [0, 5], [1, 2], [1, 4], [2, 3], [3, 4], [4, 5]]. All graph vertices and edges are supplied, and exact searches are limited to n≤9. Supplied colors=[0, 1, 0, 1, 0, 1] in vertex order. Color labels are nonnegative integer palette indices; retain their actual labels. Properness checks every edge, and using palette [0,k) is separate from merely using at most k distinct labels. Give (colors, used_colors, color_count, proper, boundary).
G001 — Use the complete finite loop-free simple undirected graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [0, 5], [1, 2], [1, 4], [2, 3], [3, 4], [4, 5]]. All graph vertices and edges are supplied, and exact searches are limited to n≤9. Supplied colors=[0, 1, 0, 1, 0, 1] in vertex order. Color labels are nonnegative integer palette indices; retain their actual labels. Properness checks every edge, and using palette [0,k) is separate from merely using at most k distinct labels. Give (colors, used_colors, color_count, proper, boundary).
Step-by-step derivation
S001 · implies
Before
source operation=chromatic-proper-audit;original snapshot and mathematics retained
After
explicit ordered source output fields=[colors, used_colors, color_count, proper, boundary]
Conditions: none
Retain the historical interface marker in the original snapshot while listing ordered source fields here,without JavaScript or external lookup.Field names are not mathematical evidence,and No retains the operation's specific claim scope.
Step identifiers and dependencies
Rule: source_ordered_interface_explicit_without_javascript · Uses: G001
S002 · implies
Before
ACTUAL complete finite carrier Fin(6);ALL original canonical simple undirected edges=[[0, 1], [0, 5], [1, 2], [1, 4], [2, 3], [3, 4], [4, 5]]
After
use full original vertex/edge data;exact search domain n<=9;empty carrier and isolated vertices RETAINED
Conditions: none
Retain every original vertex and edge.This family includes n=0 rather than importing a previous nonempty-carrier guard or deleting isolates.Actual palette labels must not be replaced by semantic relabeling.
Step identifiers and dependencies
Rule: chromatic_complete_graph_not_semantic_color_relabeling · Uses: G001, S001
S003 · implies
Before
original edge index=0;actual endpoint vertices=[0, 1];ORIGINAL palette labels=[0, 1]
After
endpoint labels different=Yes
Conditions: none
Check different colors on every original edge.Numeric labels distinguish colors,not the number of colors.One failed supplied coloring does not establish nonexistence of another proper coloring.
Step identifiers and dependencies
Rule: chromatic_every_original_edge_properness · Uses: G001, S002
S004 · implies
Before
original edge index=1;actual endpoint vertices=[0, 5];ORIGINAL palette labels=[0, 1]
After
endpoint labels different=Yes
Conditions: none
Check different colors on every original edge.Numeric labels distinguish colors,not the number of colors.One failed supplied coloring does not establish nonexistence of another proper coloring.
Step identifiers and dependencies
Rule: chromatic_every_original_edge_properness · Uses: G001, S003
S005 · implies
Before
original edge index=2;actual endpoint vertices=[1, 2];ORIGINAL palette labels=[1, 0]
After
endpoint labels different=Yes
Conditions: none
Check different colors on every original edge.Numeric labels distinguish colors,not the number of colors.One failed supplied coloring does not establish nonexistence of another proper coloring.
Step identifiers and dependencies
Rule: chromatic_every_original_edge_properness · Uses: G001, S004
S006 · implies
Before
original edge index=3;actual endpoint vertices=[1, 4];ORIGINAL palette labels=[1, 0]
After
endpoint labels different=Yes
Conditions: none
Check different colors on every original edge.Numeric labels distinguish colors,not the number of colors.One failed supplied coloring does not establish nonexistence of another proper coloring.
Step identifiers and dependencies
Rule: chromatic_every_original_edge_properness · Uses: G001, S005
S007 · implies
Before
original edge index=4;actual endpoint vertices=[2, 3];ORIGINAL palette labels=[0, 1]
After
endpoint labels different=Yes
Conditions: none
Check different colors on every original edge.Numeric labels distinguish colors,not the number of colors.One failed supplied coloring does not establish nonexistence of another proper coloring.
Step identifiers and dependencies
Rule: chromatic_every_original_edge_properness · Uses: G001, S006
S008 · implies
Before
original edge index=5;actual endpoint vertices=[3, 4];ORIGINAL palette labels=[1, 0]
After
endpoint labels different=Yes
Conditions: none
Check different colors on every original edge.Numeric labels distinguish colors,not the number of colors.One failed supplied coloring does not establish nonexistence of another proper coloring.
Step identifiers and dependencies
Rule: chromatic_every_original_edge_properness · Uses: G001, S007
S009 · implies
Before
original edge index=6;actual endpoint vertices=[4, 5];ORIGINAL palette labels=[0, 1]
After
endpoint labels different=Yes
Conditions: none
Check different colors on every original edge.Numeric labels distinguish colors,not the number of colors.One failed supplied coloring does not establish nonexistence of another proper coloring.
Step identifiers and dependencies
Rule: chromatic_every_original_edge_properness · Uses: G001, S008
S010 · calculation
Before
chromatic distinct used label count := sum(1, 1)
After
chromatic distinct used label count = 2
Conditions: none
Substitute original roles and established intermediates exactly,retaining parameter,outcome,target and denominator roles without replacing the claim.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S009
S011 · implies
Before
ORIGINAL complete color array=[0, 1, 0, 1, 0, 1];ALL edge properness flags=[Yes, Yes, Yes, Yes, Yes, Yes, Yes]
After
increasingly ordered DISTINCT used labels=[0, 1];distinct count=2;proper=Yes;not max_label+1
Conditions: none
Count distinct labels rather than max_label+1,with0 for an empty array.A proper supplied coloring gives an upper bound,not minimum status from greediness or small labels.
Step identifiers and dependencies
Rule: chromatic_distinct_label_count_not_palette_range_or_minimum · Uses: G001, S010
Result
([0, 1, 0, 1, 0, 1], [0, 1], 2, Yes, proper_coloring_only)
The full substitutions and premise checks give the original answer; source identity,format and status are unchanged.