Paper:
- Lemmas 3.4 (exactly one match) and 3.5 (all-distinct exists for 4-colourable
G) replace the earlier conjecture; both have proofs.
- Add Conjecture 3.6: every proper 3-edge-colouring of a counterexample's
reduced dual has a face with two same-colour edges that share a Kempe
cycle with the merged edge, neither of them being the merged edge.
Experiments (all under experiments/):
- search_conj_3_6_counterexample.py: finds n=14 tri#1 i_red=0 where the
algorithm's phi_t* sits in a Kempe class with no all-distinct colouring
(disproves an earlier formulation).
- check_kempe_class.py / check_kempe_class_invariance.py /
check_kempe_class_monotone.py: Kempe-class counts on H_1 and H_t* for
small triangulations; neither monotonicity direction holds.
- check_all_distinct_exists.py: even in the conj-3.6 disproof case, H_t*
itself admits all-distinct colourings in the *other* Kempe class.
- check_constrained_feasibility.py: literal H_t*-interpretation of
C1 + K0 + K1 is empirically unsatisfiable (gap in proof strategy noted).
- check_conj_face_kempe.py / check_conj_face_kempe_n15.py: test Conj 3.6
on chord-apex+Kempe colourings of reduced duals at n=12, 14, 15;
216/216 colourings on n=14 satisfy the conjecture, others vacuous.
- draw_step1_conj36.py: figure showing a Conj 3.6 witness on H_1 with two
new vertices on the witness edges and a new red bridge between them.
- draw_step1_conj36_recolored.py: same but with the Kempe cycle recoloured
alternately from merged so propriety holds.
- draw_lift_to_Gprime.py: lifts the modified+recoloured H_1 back to a
proper 3-edge-colouring of the modified G' (24+2 vertices, 39 edges,
same Tutte layout as figure 3's first graphic so positions line up).
Co-Authored-By: Claude Opus 4.7 <noreply@anthropic.com>
- Replace the dodecahedron trace at the end of section 3 with the n=14
triangulation found by search_kempe_property.py: its H_1 admits a
proper 3-edge-colouring satisfying both chord-apex and Kempe-cycle
conditions (Lemmas 2.6, 2.7).
- experiments/draw_iterated_reduction_n14.py: rebuilds fig_alg_step{0,1,2}
with Tutte barycentric layouts (outer face chosen to keep v_n in the
interior); also runs the algorithm to completion, checking chord-apex +
Kempe at each step (step 1 satisfies all; step 2 fails chord-apex;
step 3 terminates).
- Add Conjecture 3.4: G is a minimal counterexample iff no proper
3-edge-colouring of the final reduced graph H_{t*} has all (spike_t,
merged_t) pairs in distinct colours.
Co-Authored-By: Claude Opus 4.7 <noreply@anthropic.com>
- Add section 3 with Algorithm 3.1 (iterated reduction with protected edges)
and remarks on invariants and chord-apex applicability.
- Add fig:iterated-reduction-trace illustrating the algorithm on G' =
dodecahedron (G' -> H_1 -> H_2 -> terminate).
- experiments/iterated_reduction.py: Sage implementation of the algorithm.
- experiments/draw_iterated_reduction.py: produces the 3 trace figures.
- experiments/check_dodecahedron_kempe.py: enumerate proper 3-edge-colorings
of the dodecahedron's reduced dual and check the chord-apex + Kempe-cycle
conditions (0 of 36 colorings satisfy all three).
- experiments/search_kempe_property.py: search across min-deg-5
triangulations; the n = 14 first plantri triangulation is the smallest hit
(reduced dual has 20 v, 30 e).
Co-Authored-By: Claude Opus 4.7 <noreply@anthropic.com>
- Replace the chord-apex TODO with a full proof by contradiction: assume
merged != spike, define X, Y, Z, W, lift to G' so that the externals
inherit \psi(f) = (X, Y, Z, W, W), and split on W in {X, Z}. Either case
meets the hypothesis of lem:pentagonal-externals, which extends \psi to a
proper 3-edge-colouring of G' --- contradicting non-3-edge-colourability
via Tait.
- Add fig:chord-apex-proof: the assumed reduced-dual colouring on top, and
the two lifted-G' cases (W=Z, W=X) below, rendered on the dodecahedron.
- Add experiments/draw_chord_apex_proof.py.
Co-Authored-By: Claude Opus 4.7 <noreply@anthropic.com>
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>
didericis