Previous figures drew σ as VERTEX colors on a 6-cycle. This was
misleading: σ = (1,2,3,2,3,1) is not a proper vertex 3-coloring of
a hexagon (σ_5 = σ_0 = 1 at adjacent vertices), and the user
correctly flagged this.
σ is the coloring of the 6 *spoke edges* -- the G'-edges of G' that
cross γ, equivalently the 6 edges of γ ⊂ G under the duality
γ-edge ↔ crossing G'-edge.
Adjacent γ-edges meet at γ-vertices, which are not G'-vertices, so
σ has NO proper-coloring constraint on itself. Proper-edge-coloring
constraints live on each tire's full annular cycle, which is longer
than 6 (T_1's is 12, T_2's is 9), with γ-spokes interleaved among
non-γ spokes; that's where the extendibility of σ is actually
checked.
Redrawn figures:
- fig_rainbow_orbit.png: σ drawn as edge colors of γ (not vertex
colors), all 6 orbit elements.
- fig_rainbow_pattern.png: abstract pattern abcbca as edge labels,
with explanatory text in the legend.
- fig_rainbow_setup.png: shows γ between the two tires with each
tire's full annular cycle (length 12 and 9), the interleaved
non-γ dual vertices, the dashed G'-spoke edges crossing γ
colored by σ, and T_1's antipodal chord in O_1.
Co-Authored-By: Claude Opus 4.7 <noreply@anthropic.com>
didericis
8dd9d537f9