Define Kempe-balanced colourings of a full medial tire graph (Def 5.7):
for each valid face (outer face or interior non-tooth face of B(T)) and
each colour pair {a,b}, the number of tooth apexes incident to the face
coloured a or b must be even. Add Remark 5.8 (necessity: a colouring of
M(T) extends to M(G) only if it is Kempe-balanced) and rename Lemma 5.5
to "Kempe chains are cycles".
Add kempe_valid_colorings.py: enumerate all proper 3-colourings of a full
medial tire graph, label each Kempe-balanced/valid or invalid, and plot
them with the offending face's Kempe chains and odd apex set highlighted
on invalid panels.
Co-Authored-By: Claude Opus 4.8 <noreply@anthropic.com>
Name A(T) the "annular cycle" (Thm 3.3, Def 3.4); clarify the bite-face
condition in Remark 3.8 to count down-tooth apexes interior to each face;
add the non-incidence stipulation for bite edges to Def 3.7.
Add an exhaustive generator over |A(T)| enforcing the 3.1-3.9 properties
(tooth word, non-crossing non-incident bites, >=3 up teeth, bite-face
condition), plus a plotting script and the n=9 atlas (81 dihedral classes).
Co-Authored-By: Claude Opus 4.8 <noreply@anthropic.com>
Adds two TikZ figures (boundary-state worst cases and annular cycle
counterexample), a new subsection on Kempe-cycle conservation across
medial tires, and the experiment scripts/findings for the medial tire
restriction search and annular cycle condition check.
Co-Authored-By: Claude Sonnet 4.6 <noreply@anthropic.com>
didericis