For a random 12-vertex maximal planar graph (sphere convex hull), enumerate
all proper 3-colourings of M(G), take the BFS-level (tire-tree) decomposition
from every source vertex, and build each full medial tire graph M(T) in the
ambient tread-face model (cycle + teeth + bites). Recognise each M(T) as a
FullMedialTireGraph and label every proper 3-colouring Realized (Kempe-balanced
and a restriction of a global colouring), Unrealized (balanced but not a
restriction), or Invalid (not balanced).
Findings on seed 1 (17 pieces, M(G) with 90 colourings): zero realized-but-
invalid colourings (confirms Remark 5.8 on a real triangulation), and 12 of 17
pieces carry Unrealized colourings -- Kempe-balance is necessary but not
sufficient for realization; it is sufficient only on cap-like all-up/shallow
treads.
Co-Authored-By: Claude Opus 4.8 <noreply@anthropic.com>
didericis
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