Introduce Conjecture 1.31 (tire inner-boundary three-colour) as a
decomposition-native weakening of the level-cycle conjecture 1.29:
every maximal planar graph admits a vertex source and proper 4-colouring
under which each tire inner boundary omits a colour. Remark 1.32 shows
inner boundaries are single-level cycles, so the vertex-source form of
1.29 implies it on 2-connected boundaries.
Extend check_level_cycle_three_color.py with --restriction inner-boundary
(reconstructs the tire-tree decomposition from the embedding; inner
boundary = level-(d+1) vertices of each depth-d dual component) and a
--min-connectivity flag for the 5-connected slice.
Verified: full census 4<=n<=13 (57716 triangulations) and 5-connected
slice 14<=n<=24 (9732 graphs) all admit witnesses; no counterexample.
Co-Authored-By: Claude Opus 4.8 <noreply@anthropic.com>
didericis
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