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Add Kempe-balanced colouring definition and validity classifier

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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>
This commit is contained in:
didericis
2026-06-11 16:00:10 -04:00
parent 8cc94fb6b9
commit 79cbca8e00
8 changed files with 427 additions and 10 deletions
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papers/medial_tire_decompositions_of_plane_triangulations/paper.log
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This is pdfTeX, Version 3.141592653-2.6-1.40.24 (TeX Live 2022) (preloaded format=pdflatex 2022.10.5) 11 JUN 2026 15:59
entering extended mode
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[4] [5] [6] [7] [8] [9] [10] (./paper.aux) )
Here is how much of TeX's memory you used:
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s/cm/cmti8.pfb></usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/
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Output written on paper.pdf (10 pages, 272876 bytes).
Output written on paper.pdf (10 pages, 276272 bytes).
PDF statistics:
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