Step-by-step derivation
S001 · implies
Before
source operation=chromatic-minimum-boundary;original snapshot and mathematics retained
After
explicit ordered source output fields=[k, candidate_k_palette_valid, chromatic_number, k_equals_chromatic_number, 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, 2], [0, 3], [0, 5], [1, 2], [2, 3], [2, 5], [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, 2];ORIGINAL palette labels=[0, 2]
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=[0, 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, S004
S006 · implies
Before
original edge index=3;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, S005
S007 · implies
Before
original edge index=4;actual endpoint vertices=[1, 2];ORIGINAL palette labels=[1, 2]
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=[2, 3];ORIGINAL palette labels=[2, 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, S007
S009 · implies
Before
original edge index=6;actual endpoint vertices=[2, 5];ORIGINAL palette labels=[2, 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 · implies
Before
original edge index=7;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, S009
S011 · implies
Before
original edge index=8;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, S010
S012 · calculation
Before
chromatic distinct used label count := sum(1, 1, 1)
After
chromatic distinct used label count = 3
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, S011
S013 · implies
Before
ORIGINAL complete color array=[0, 1, 2, 1, 0, 1];ALL edge properness flags=[Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes]
After
increasingly ordered DISTINCT used labels=[0, 1, 2];distinct count=3;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, S012
S014 · implies
Before
requested ACTUAL palette[0,4);ORIGINAL labels=[0, 1, 2, 1, 0, 1];per-vertex label<k flags=[Yes, Yes, Yes, Yes, Yes, Yes];proper=Yes
After
within requested palette=Yes;THIS candidate upper certificate=Yes;distinct used-label count=3 is a DIFFERENT test
Conditions: none
Even if at most k distinct labels are used,the supplied palette certificate fails when original labels lie outside[0,k).Semantic relabeling must not replace source data.Failure rejects this candidate,not existence of another k-coloring.
Step identifiers and dependencies
Rule: chromatic_original_palette_range_not_semantic_renaming · Uses: G001, S013
S015 · implies
Before
ACTUAL graph Fin(6);try palette[0,1);original vertex order0..5
After
search ALL earlier lexicographic prefixes:vertex order fixed,each color tried increasingly;only an actual earlier-edge conflict may prune
Conditions: none
Search each k completely rather than treating a failed greedy run as non-colorability.Fixed original vertex order and increasing colors define the source's first lexicographic witness.Prune only an actual edge conflict;otherwise continue or exhaustively backtrack.
Step identifiers and dependencies
Rule: chromatic_complete_lex_search_not_greedy_optimality · Uses: G001, S014
S016 · implies
Before
palette=1;original vertex=0;assigned ORIGINAL prefix=[];try next ascending color=0
After
ALL earlier adjacent vertices already using this label=[];admissible extension=Yes
Conditions: none
List all actual earlier-neighbor conflicts for this prefix/color.A conflict rejects every completion of that extension;otherwise recursively check all remaining vertices rather than calling a local pass a complete solution.
Step identifiers and dependencies
Rule: chromatic_every_prefix_color_conflict_or_recursion · Uses: G001, S015
S017 · implies
Before
palette=1;original vertex=1;assigned ORIGINAL prefix=[0];try next ascending color=0
After
ALL earlier adjacent vertices already using this label=[0];admissible extension=No
Conditions: none
List all actual earlier-neighbor conflicts for this prefix/color.A conflict rejects every completion of that extension;otherwise recursively check all remaining vertices rather than calling a local pass a complete solution.
Step identifiers and dependencies
Rule: chromatic_every_prefix_color_conflict_or_recursion · Uses: G001, S016
S018 · implies
Before
palette=1;original vertex=1;original fixed prefix=[0];ALL colors0..0 attempted
After
no valid completion of this prefix in this palette
Conditions: none
All allowed labels at this vertex have been tried,so this fixed prefix has no completion.Only this conclusion at the root excludes the whole k-palette.
Step identifiers and dependencies
Rule: chromatic_full_palette_exhaustion_at_prefix · Uses: G001, S017
S019 · implies
Before
palette=1;original prefix=[];vertex=0;chosen color=0;ALL its later completions exhausted
After
restore this unassigned coordinate and try the next ascending color
Conditions: none
Backtrack only after the whole subtree fails.Every later completion of this branch was excluded by an actual conflict or a completely exhausted subtree,not one sampled attempt.
Step identifiers and dependencies
Rule: chromatic_exhausted_branch_backtracking_not_one_bad_sample · Uses: G001, S018
S020 · implies
Before
palette=1;original vertex=0;original fixed prefix=[];ALL colors0..0 attempted
After
no valid completion of this prefix in this palette
Conditions: none
All allowed labels at this vertex have been tried,so this fixed prefix has no completion.Only this conclusion at the root excludes the whole k-palette.
Step identifiers and dependencies
Rule: chromatic_full_palette_exhaustion_at_prefix · Uses: G001, S019
S021 · implies
Before
palette[0,1) searched from EMPTY root prefix with full branch exclusions
After
NO proper coloring exists in this palette;any smaller palette is a subset of it
Conditions: none
Exhausting the complete root tree supplies the lower certificate.Any smaller palette embeds in this one and is excluded as well;no greedy path establishes this conclusion.
Step identifiers and dependencies
Rule: chromatic_actual_lower_palette_exhaustion · Uses: G001, S020
S022 · implies
Before
ACTUAL graph Fin(6);try palette[0,2);original vertex order0..5
After
search ALL earlier lexicographic prefixes:vertex order fixed,each color tried increasingly;only an actual earlier-edge conflict may prune
Conditions: none
Search each k completely rather than treating a failed greedy run as non-colorability.Fixed original vertex order and increasing colors define the source's first lexicographic witness.Prune only an actual edge conflict;otherwise continue or exhaustively backtrack.
Step identifiers and dependencies
Rule: chromatic_complete_lex_search_not_greedy_optimality · Uses: G001, S021
S023 · implies
Before
palette=2;original vertex=0;assigned ORIGINAL prefix=[];try next ascending color=0
After
ALL earlier adjacent vertices already using this label=[];admissible extension=Yes
Conditions: none
List all actual earlier-neighbor conflicts for this prefix/color.A conflict rejects every completion of that extension;otherwise recursively check all remaining vertices rather than calling a local pass a complete solution.
Step identifiers and dependencies
Rule: chromatic_every_prefix_color_conflict_or_recursion · Uses: G001, S022
S024 · implies
Before
palette=2;original vertex=1;assigned ORIGINAL prefix=[0];try next ascending color=0
After
ALL earlier adjacent vertices already using this label=[0];admissible extension=No
Conditions: none
List all actual earlier-neighbor conflicts for this prefix/color.A conflict rejects every completion of that extension;otherwise recursively check all remaining vertices rather than calling a local pass a complete solution.
Step identifiers and dependencies
Rule: chromatic_every_prefix_color_conflict_or_recursion · Uses: G001, S023
S025 · implies
Before
palette=2;original vertex=1;assigned ORIGINAL prefix=[0];try next ascending color=1
After
ALL earlier adjacent vertices already using this label=[];admissible extension=Yes
Conditions: none
List all actual earlier-neighbor conflicts for this prefix/color.A conflict rejects every completion of that extension;otherwise recursively check all remaining vertices rather than calling a local pass a complete solution.
Step identifiers and dependencies
Rule: chromatic_every_prefix_color_conflict_or_recursion · Uses: G001, S024
S026 · implies
Before
palette=2;original vertex=2;assigned ORIGINAL prefix=[0, 1];try next ascending color=0
After
ALL earlier adjacent vertices already using this label=[0];admissible extension=No
Conditions: none
List all actual earlier-neighbor conflicts for this prefix/color.A conflict rejects every completion of that extension;otherwise recursively check all remaining vertices rather than calling a local pass a complete solution.
Step identifiers and dependencies
Rule: chromatic_every_prefix_color_conflict_or_recursion · Uses: G001, S025
S027 · implies
Before
palette=2;original vertex=2;assigned ORIGINAL prefix=[0, 1];try next ascending color=1
After
ALL earlier adjacent vertices already using this label=[1];admissible extension=No
Conditions: none
List all actual earlier-neighbor conflicts for this prefix/color.A conflict rejects every completion of that extension;otherwise recursively check all remaining vertices rather than calling a local pass a complete solution.
Step identifiers and dependencies
Rule: chromatic_every_prefix_color_conflict_or_recursion · Uses: G001, S026
S028 · implies
Before
palette=2;original vertex=2;original fixed prefix=[0, 1];ALL colors0..1 attempted
After
no valid completion of this prefix in this palette
Conditions: none
All allowed labels at this vertex have been tried,so this fixed prefix has no completion.Only this conclusion at the root excludes the whole k-palette.
Step identifiers and dependencies
Rule: chromatic_full_palette_exhaustion_at_prefix · Uses: G001, S027
S029 · implies
Before
palette=2;original prefix=[0];vertex=1;chosen color=1;ALL its later completions exhausted
After
restore this unassigned coordinate and try the next ascending color
Conditions: none
Backtrack only after the whole subtree fails.Every later completion of this branch was excluded by an actual conflict or a completely exhausted subtree,not one sampled attempt.
Step identifiers and dependencies
Rule: chromatic_exhausted_branch_backtracking_not_one_bad_sample · Uses: G001, S028
S030 · implies
Before
palette=2;original vertex=1;original fixed prefix=[0];ALL colors0..1 attempted
After
no valid completion of this prefix in this palette
Conditions: none
All allowed labels at this vertex have been tried,so this fixed prefix has no completion.Only this conclusion at the root excludes the whole k-palette.
Step identifiers and dependencies
Rule: chromatic_full_palette_exhaustion_at_prefix · Uses: G001, S029
S031 · implies
Before
palette=2;original prefix=[];vertex=0;chosen color=0;ALL its later completions exhausted
After
restore this unassigned coordinate and try the next ascending color
Conditions: none
Backtrack only after the whole subtree fails.Every later completion of this branch was excluded by an actual conflict or a completely exhausted subtree,not one sampled attempt.
Step identifiers and dependencies
Rule: chromatic_exhausted_branch_backtracking_not_one_bad_sample · Uses: G001, S030
S032 · implies
Before
palette=2;original vertex=0;assigned ORIGINAL prefix=[];try next ascending color=1
After
ALL earlier adjacent vertices already using this label=[];admissible extension=Yes
Conditions: none
List all actual earlier-neighbor conflicts for this prefix/color.A conflict rejects every completion of that extension;otherwise recursively check all remaining vertices rather than calling a local pass a complete solution.
Step identifiers and dependencies
Rule: chromatic_every_prefix_color_conflict_or_recursion · Uses: G001, S031
S033 · implies
Before
palette=2;original vertex=1;assigned ORIGINAL prefix=[1];try next ascending color=0
After
ALL earlier adjacent vertices already using this label=[];admissible extension=Yes
Conditions: none
List all actual earlier-neighbor conflicts for this prefix/color.A conflict rejects every completion of that extension;otherwise recursively check all remaining vertices rather than calling a local pass a complete solution.
Step identifiers and dependencies
Rule: chromatic_every_prefix_color_conflict_or_recursion · Uses: G001, S032
S034 · implies
Before
palette=2;original vertex=2;assigned ORIGINAL prefix=[1, 0];try next ascending color=0
After
ALL earlier adjacent vertices already using this label=[1];admissible extension=No
Conditions: none
List all actual earlier-neighbor conflicts for this prefix/color.A conflict rejects every completion of that extension;otherwise recursively check all remaining vertices rather than calling a local pass a complete solution.
Step identifiers and dependencies
Rule: chromatic_every_prefix_color_conflict_or_recursion · Uses: G001, S033
S035 · implies
Before
palette=2;original vertex=2;assigned ORIGINAL prefix=[1, 0];try next ascending color=1
After
ALL earlier adjacent vertices already using this label=[0];admissible extension=No
Conditions: none
List all actual earlier-neighbor conflicts for this prefix/color.A conflict rejects every completion of that extension;otherwise recursively check all remaining vertices rather than calling a local pass a complete solution.
Step identifiers and dependencies
Rule: chromatic_every_prefix_color_conflict_or_recursion · Uses: G001, S034
S036 · implies
Before
palette=2;original vertex=2;original fixed prefix=[1, 0];ALL colors0..1 attempted
After
no valid completion of this prefix in this palette
Conditions: none
All allowed labels at this vertex have been tried,so this fixed prefix has no completion.Only this conclusion at the root excludes the whole k-palette.
Step identifiers and dependencies
Rule: chromatic_full_palette_exhaustion_at_prefix · Uses: G001, S035
S037 · implies
Before
palette=2;original prefix=[1];vertex=1;chosen color=0;ALL its later completions exhausted
After
restore this unassigned coordinate and try the next ascending color
Conditions: none
Backtrack only after the whole subtree fails.Every later completion of this branch was excluded by an actual conflict or a completely exhausted subtree,not one sampled attempt.
Step identifiers and dependencies
Rule: chromatic_exhausted_branch_backtracking_not_one_bad_sample · Uses: G001, S036
S038 · implies
Before
palette=2;original vertex=1;assigned ORIGINAL prefix=[1];try next ascending color=1
After
ALL earlier adjacent vertices already using this label=[0];admissible extension=No
Conditions: none
List all actual earlier-neighbor conflicts for this prefix/color.A conflict rejects every completion of that extension;otherwise recursively check all remaining vertices rather than calling a local pass a complete solution.
Step identifiers and dependencies
Rule: chromatic_every_prefix_color_conflict_or_recursion · Uses: G001, S037
S039 · implies
Before
palette=2;original vertex=1;original fixed prefix=[1];ALL colors0..1 attempted
After
no valid completion of this prefix in this palette
Conditions: none
All allowed labels at this vertex have been tried,so this fixed prefix has no completion.Only this conclusion at the root excludes the whole k-palette.
Step identifiers and dependencies
Rule: chromatic_full_palette_exhaustion_at_prefix · Uses: G001, S038
S040 · implies
Before
palette=2;original prefix=[];vertex=0;chosen color=1;ALL its later completions exhausted
After
restore this unassigned coordinate and try the next ascending color
Conditions: none
Backtrack only after the whole subtree fails.Every later completion of this branch was excluded by an actual conflict or a completely exhausted subtree,not one sampled attempt.
Step identifiers and dependencies
Rule: chromatic_exhausted_branch_backtracking_not_one_bad_sample · Uses: G001, S039
S041 · implies
Before
palette=2;original vertex=0;original fixed prefix=[];ALL colors0..1 attempted
After
no valid completion of this prefix in this palette
Conditions: none
All allowed labels at this vertex have been tried,so this fixed prefix has no completion.Only this conclusion at the root excludes the whole k-palette.
Step identifiers and dependencies
Rule: chromatic_full_palette_exhaustion_at_prefix · Uses: G001, S040
S042 · implies
Before
palette[0,2) searched from EMPTY root prefix with full branch exclusions
After
NO proper coloring exists in this palette;any smaller palette is a subset of it
Conditions: none
Exhausting the complete root tree supplies the lower certificate.Any smaller palette embeds in this one and is excluded as well;no greedy path establishes this conclusion.
Step identifiers and dependencies
Rule: chromatic_actual_lower_palette_exhaustion · Uses: G001, S041
S043 · implies
Before
ACTUAL graph Fin(6);try palette[0,3);original vertex order0..5
After
search ALL earlier lexicographic prefixes:vertex order fixed,each color tried increasingly;only an actual earlier-edge conflict may prune
Conditions: none
Search each k completely rather than treating a failed greedy run as non-colorability.Fixed original vertex order and increasing colors define the source's first lexicographic witness.Prune only an actual edge conflict;otherwise continue or exhaustively backtrack.
Step identifiers and dependencies
Rule: chromatic_complete_lex_search_not_greedy_optimality · Uses: G001, S042
S044 · implies
Before
palette=3;original vertex=0;assigned ORIGINAL prefix=[];try next ascending color=0
After
ALL earlier adjacent vertices already using this label=[];admissible extension=Yes
Conditions: none
List all actual earlier-neighbor conflicts for this prefix/color.A conflict rejects every completion of that extension;otherwise recursively check all remaining vertices rather than calling a local pass a complete solution.
Step identifiers and dependencies
Rule: chromatic_every_prefix_color_conflict_or_recursion · Uses: G001, S043
S045 · implies
Before
palette=3;original vertex=1;assigned ORIGINAL prefix=[0];try next ascending color=0
After
ALL earlier adjacent vertices already using this label=[0];admissible extension=No
Conditions: none
List all actual earlier-neighbor conflicts for this prefix/color.A conflict rejects every completion of that extension;otherwise recursively check all remaining vertices rather than calling a local pass a complete solution.
Step identifiers and dependencies
Rule: chromatic_every_prefix_color_conflict_or_recursion · Uses: G001, S044
S046 · implies
Before
palette=3;original vertex=1;assigned ORIGINAL prefix=[0];try next ascending color=1
After
ALL earlier adjacent vertices already using this label=[];admissible extension=Yes
Conditions: none
List all actual earlier-neighbor conflicts for this prefix/color.A conflict rejects every completion of that extension;otherwise recursively check all remaining vertices rather than calling a local pass a complete solution.
Step identifiers and dependencies
Rule: chromatic_every_prefix_color_conflict_or_recursion · Uses: G001, S045
S047 · implies
Before
palette=3;original vertex=2;assigned ORIGINAL prefix=[0, 1];try next ascending color=0
After
ALL earlier adjacent vertices already using this label=[0];admissible extension=No
Conditions: none
List all actual earlier-neighbor conflicts for this prefix/color.A conflict rejects every completion of that extension;otherwise recursively check all remaining vertices rather than calling a local pass a complete solution.
Step identifiers and dependencies
Rule: chromatic_every_prefix_color_conflict_or_recursion · Uses: G001, S046
S048 · implies
Before
palette=3;original vertex=2;assigned ORIGINAL prefix=[0, 1];try next ascending color=1
After
ALL earlier adjacent vertices already using this label=[1];admissible extension=No
Conditions: none
List all actual earlier-neighbor conflicts for this prefix/color.A conflict rejects every completion of that extension;otherwise recursively check all remaining vertices rather than calling a local pass a complete solution.
Step identifiers and dependencies
Rule: chromatic_every_prefix_color_conflict_or_recursion · Uses: G001, S047
S049 · implies
Before
palette=3;original vertex=2;assigned ORIGINAL prefix=[0, 1];try next ascending color=2
After
ALL earlier adjacent vertices already using this label=[];admissible extension=Yes
Conditions: none
List all actual earlier-neighbor conflicts for this prefix/color.A conflict rejects every completion of that extension;otherwise recursively check all remaining vertices rather than calling a local pass a complete solution.
Step identifiers and dependencies
Rule: chromatic_every_prefix_color_conflict_or_recursion · Uses: G001, S048
S050 · implies
Before
palette=3;original vertex=3;assigned ORIGINAL prefix=[0, 1, 2];try next ascending color=0
After
ALL earlier adjacent vertices already using this label=[0];admissible extension=No
Conditions: none
List all actual earlier-neighbor conflicts for this prefix/color.A conflict rejects every completion of that extension;otherwise recursively check all remaining vertices rather than calling a local pass a complete solution.
Step identifiers and dependencies
Rule: chromatic_every_prefix_color_conflict_or_recursion · Uses: G001, S049
S051 · implies
Before
palette=3;original vertex=3;assigned ORIGINAL prefix=[0, 1, 2];try next ascending color=1
After
ALL earlier adjacent vertices already using this label=[];admissible extension=Yes
Conditions: none
List all actual earlier-neighbor conflicts for this prefix/color.A conflict rejects every completion of that extension;otherwise recursively check all remaining vertices rather than calling a local pass a complete solution.
Step identifiers and dependencies
Rule: chromatic_every_prefix_color_conflict_or_recursion · Uses: G001, S050
S052 · implies
Before
palette=3;original vertex=4;assigned ORIGINAL prefix=[0, 1, 2, 1];try next ascending color=0
After
ALL earlier adjacent vertices already using this label=[];admissible extension=Yes
Conditions: none
List all actual earlier-neighbor conflicts for this prefix/color.A conflict rejects every completion of that extension;otherwise recursively check all remaining vertices rather than calling a local pass a complete solution.
Step identifiers and dependencies
Rule: chromatic_every_prefix_color_conflict_or_recursion · Uses: G001, S051
S053 · implies
Before
palette=3;original vertex=5;assigned ORIGINAL prefix=[0, 1, 2, 1, 0];try next ascending color=0
After
ALL earlier adjacent vertices already using this label=[0, 4];admissible extension=No
Conditions: none
List all actual earlier-neighbor conflicts for this prefix/color.A conflict rejects every completion of that extension;otherwise recursively check all remaining vertices rather than calling a local pass a complete solution.
Step identifiers and dependencies
Rule: chromatic_every_prefix_color_conflict_or_recursion · Uses: G001, S052
S054 · implies
Before
palette=3;original vertex=5;assigned ORIGINAL prefix=[0, 1, 2, 1, 0];try next ascending color=1
After
ALL earlier adjacent vertices already using this label=[];admissible extension=Yes
Conditions: none
List all actual earlier-neighbor conflicts for this prefix/color.A conflict rejects every completion of that extension;otherwise recursively check all remaining vertices rather than calling a local pass a complete solution.
Step identifiers and dependencies
Rule: chromatic_every_prefix_color_conflict_or_recursion · Uses: G001, S053
S055 · implies
Before
FIRST completed vertex-order increasing-color search witness=[0, 1, 2, 1, 0, 1];palette size=3;ALL smaller palettes exhausted
After
every ORIGINAL edge endpoints different;constructive chi<= 3 and exhaustive chi>= 3;chi=3;first LEX witness retained
Conditions: none
The complete proper original-edge coloring gives the upper bound;the preceding exhaustive failures give the lower bound.Their equality establishes the exact minimum.Fixed DFS order preserves the source's first lexicographic witness,not an arbitrary optimum.A nonempty edgeless graph succeeds first at k=1 with all zeros;no fictitious0-color search is needed.
Step identifiers and dependencies
Rule: chromatic_constructive_upper_and_exhausted_lower_tie_order · Uses: G001, S054
S056 · implies
Before
requested k=4;THIS supplied candidate validity=Yes;SEPARATE exact chi=3;SEPARATE canonical optimum witness=[0, 1, 2, 1, 0, 1]
After
k equals chromatic number=No;comparison does NOT certify supplied colors;at-most-k and exact-k-minimum are separate
Conditions: none
k may equal the true chromatic number while the supplied coloring fails.Or a valid k-palette may only give chi≤k with equality false.Keep these separate questions and do not copy one successful flag into another.
Step identifiers and dependencies
Rule: chromatic_k_equals_chi_independent_of_supplied_candidate · Uses: G001, S055