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], [4, 5]]. 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. Total vertex colors=[0, 1, 0, 1, 0, 1] are labels, not cardinalities. Only adjacent endpoints must differ; nonadjacent vertices may share a color. A proper k-coloring uses at most k distinct labels, even if their numerical names exceed k−1. One such coloring gives only a chromatic-number upper bound, not optimality or a general greedy guarantee. Color relabel table=[24, 54] is indexed by the original color labels and must be injective on the labels actually used. Give (colors,recolored,proper_before?,proper_after?,same_conflicts?,color_names_do_not_change_vertex_partition).
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], [4, 5]]. 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. Total vertex colors=[0, 1, 0, 1, 0, 1] are labels, not cardinalities. Only adjacent endpoints must differ; nonadjacent vertices may share a color. A proper k-coloring uses at most k distinct labels, even if their numerical names exceed k−1. One such coloring gives only a chromatic-number upper bound, not optimality or a general greedy guarantee. Color relabel table=[24, 54] is indexed by the original color labels and must be injective on the labels actually used. Give (colors,recolored,proper_before?,proper_after?,same_conflicts?,color_names_do_not_change_vertex_partition).
Step-by-step derivation
S001 · implies
Before
complete undirected vertex carrier Fin(6);original edges=[[0, 1], [1, 2], [2, 3], [3, 4], [4, 5]]
After
normalized sorted simple edges=[[0, 1], [1, 2], [2, 3], [3, 4], [4, 5]];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 dependencies
Rule: graph_complete_simple_carrier_normalized_edge_order · Uses: G001
S002 · implies
Before
colors:normalized edge index=0;edge=[0, 1];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 dependencies
Rule: graph_every_edge_endpoint_constraint · Uses: G001, S001
S003 · implies
Before
colors:normalized edge index=1;edge=[1, 2];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 dependencies
Rule: graph_every_edge_endpoint_constraint · Uses: G001, S002
S004 · implies
Before
colors:normalized edge index=2;edge=[2, 3];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 dependencies
Rule: graph_every_edge_endpoint_constraint · Uses: G001, S003
S005 · implies
Before
colors:normalized edge index=3;edge=[3, 4];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 dependencies
Rule: graph_every_edge_endpoint_constraint · Uses: G001, S004
S006 · implies
Before
colors:normalized edge index=4;edge=[4, 5];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 dependencies
Rule: graph_every_edge_endpoint_constraint · Uses: G001, S005
S007 · implies
Before
total original vertex colors=[0, 1, 0, 1, 0, 1];all normalized conflicts=[]
After
increasing distinct used labels=[0, 1];used count=2;all edges proper=Yes
Conditions: none
Colors are names:count distinct used labels rather than maximum label plus1.Only adjacent equal colors conflict;nonadjacent color reuse is allowed.
Step identifiers and dependencies
Rule: coloring_distinct_label_count_not_numeric_magnitude · Uses: G001, S006
S008 · implies
Before
original used labels=[0, 1];raw relabel table=[24, 54]
After
used-label image pairs=[[0, 24], [1, 54]];injective on USED labels=Yes;recolored original vertices=[24, 54, 24, 54, 24, 54]
Conditions: none
Require table entries and injectivity only for colors actually used,not a global permutation on unused entries.Apply the table in original vertex order,indexed by the actual color labels.
Step identifiers and dependencies
Rule: coloring_injective_used_label_renaming_not_global_permutation · Uses: G001, S007
S009 · implies
Before
normalized edge index=0;edge=[0, 1];old endpoint colors=[0, 1];new endpoint colors=[24, 54]
After
equal before=No;equal after=No;same conflict test=Yes
Conditions: none
Injective renaming preserves equality and inequality,so every edge's conflict status remains unchanged.A previously improper coloring is not repaired by mere renaming.
Step identifiers and dependencies
Rule: coloring_every_conflict_preserved_by_injective_renaming · Uses: G001, S008
S010 · implies
Before
normalized edge index=1;edge=[1, 2];old endpoint colors=[1, 0];new endpoint colors=[54, 24]
After
equal before=No;equal after=No;same conflict test=Yes
Conditions: none
Injective renaming preserves equality and inequality,so every edge's conflict status remains unchanged.A previously improper coloring is not repaired by mere renaming.
Step identifiers and dependencies
Rule: coloring_every_conflict_preserved_by_injective_renaming · Uses: G001, S009
S011 · implies
Before
normalized edge index=2;edge=[2, 3];old endpoint colors=[0, 1];new endpoint colors=[24, 54]
After
equal before=No;equal after=No;same conflict test=Yes
Conditions: none
Injective renaming preserves equality and inequality,so every edge's conflict status remains unchanged.A previously improper coloring is not repaired by mere renaming.
Step identifiers and dependencies
Rule: coloring_every_conflict_preserved_by_injective_renaming · Uses: G001, S010
S012 · implies
Before
normalized edge index=3;edge=[3, 4];old endpoint colors=[1, 0];new endpoint colors=[54, 24]
After
equal before=No;equal after=No;same conflict test=Yes
Conditions: none
Injective renaming preserves equality and inequality,so every edge's conflict status remains unchanged.A previously improper coloring is not repaired by mere renaming.
Step identifiers and dependencies
Rule: coloring_every_conflict_preserved_by_injective_renaming · Uses: G001, S011
S013 · implies
Before
normalized edge index=4;edge=[4, 5];old endpoint colors=[0, 1];new endpoint colors=[24, 54]
After
equal before=No;equal after=No;same conflict test=Yes
Conditions: none
Injective renaming preserves equality and inequality,so every edge's conflict status remains unchanged.A previously improper coloring is not repaired by mere renaming.
Step identifiers and dependencies
Rule: coloring_every_conflict_preserved_by_injective_renaming · Uses: G001, S012
Result
([0, 1, 0, 1, 0, 1], [24, 54, 24, 54, 24, 54], Yes, Yes, Yes, color_names_do_not_change_vertex_partition)
The full substitutions and premise checks give the original answer; source identity,format and status are unchanged.