Add Definition 2.1 (reduced dual) and a remark on cubicity/planarity, plus an
experiment verifying it on the icosahedron/dodecahedron and four figures, one
per construction step.
reduced_dual.py builds G' = dodecahedron (dual of the icosahedron), applies the
construction, and confirms the result is a cubic, planar, simple graph whose
dual is a simple triangulation. Finding: the construction is an n -> n-2
reduction (12 -> 10 here), not n-1, since the single apex v_n collapses one more
vertex than a standard pentagon re-triangulation; the result also re-introduces
degree-3 and degree-4 vertices (degree seq [7,5,5,5,5,5,5,4,4,3]).
draw_reduced_dual_steps.py renders fig_reduced_dual_step1..4.png, embedded as a
2x2 grid after the definition.
Co-Authored-By: Claude Opus 4.7 (1M context) <noreply@anthropic.com>
Remove the Introduction and Strategy sections and everything after the
separating-cycle definition (no-separating-triangle lemma, 5-connectivity
proposition, and the Step 2-6 stubs). Rename the section heading from
"Step 1: The minimal counterexample" to "The minimal counterexample", drop
the now-unused separating-cycle definition, and adjust the lead-in to mention
only the degree reduction. Remaining: reduction-to-triangulations lemma,
minimal-counterexample definition, |V|>=12 remark, and minimum-degree-5 lemma.
Co-Authored-By: Claude Opus 4.7 (1M context) <noreply@anthropic.com>
New paper "Dual Decomposition of Minimal Counterexamples" outlining a six-step
cut-and-recombine attack on the 4CT via the dual cubic graph: minimal
counterexample -> dualise -> minimum (cyclic) edge cut -> cap to cubic ->
3-edge-colour the pieces -> reconnect. Strategy section flags steps 1-5 as
standard machinery and step 6 (recombination) as the crux.
Step 1 written in full: reduction to triangulations, definition of the minimal
counterexample, minimum-degree >= 5 (degree <=3 and degree-4 Kempe cases), and
no separating triangle => 4-connected. 5-connectivity stated as Birkhoff's
separating-4-cycle reduction (attributed, not re-derived).
Co-Authored-By: Claude Opus 4.7 (1M context) <noreply@anthropic.com>
didericis