- New experiment experiments/check_conj_on_holton_mckay.py parses
McKay's planar_code file of the 6 non-Hamiltonian 38-vertex cubic
plane graphs (Holton-McKay) and tests both clauses (1)-(3) and (1)-(4)
of the face-monochromatic-pair conjecture on each. Result: 17,280
candidate colourings, all 17,280 satisfy both conjectures.
- Add a "Targeted check on the Holton-McKay duals" paragraph to
Remark 4.4 with a per-graph table.
- Fix a latent bug in check_conj_3_8_scaled.py: b was hardcoded to
cyc_b, leaving b == a when phi(e_1) == cyc_b (and consequently c
ambiguous). Now correctly computes b = whichever of cyc_a/cyc_b is
not a, raising if neither matches. The bug never crashed n <= 20
because any() short-circuited on correctly-built witnesses; the
Holton-McKay reductions hit it on the first witness, surfacing it.
Co-Authored-By: Claude Opus 4.7 <noreply@anthropic.com>
- Fill in the abstract and add a Section 1 introduction.
- Rename "cubic-graph edge contraction" -> "edge suppression" throughout
(section, definition, theorem, captions, prose, labels). The PNG
filenames keep their old paths and still resolve.
- Reframe edge suppression as a classical operation we recall, not a new
concept we introduce; the face-monochromatic-pair conjecture (with its
strengthening) is the sole contribution.
- Add a bibliography citing Appel-Haken Parts I and II, Robertson-Sanders-
Seymour-Thomas, and Gonthier, and \cite them in the intro.
Co-Authored-By: Claude Opus 4.7 <noreply@anthropic.com>
n=19 (21,138 col., 68s) and n=20 (107,874 col., 361s) both pass.
New total for clauses (1)-(4) over n<=20: 142,812/142,812.
Also bump max_n in check_conj_3_8_scaled.py default to 20 (was 18) and
time_budget_per_n to 7200s.
Co-Authored-By: Claude Opus 4.7 <noreply@anthropic.com>
- Move the iterated-reduction algorithm, its two structural lemmas
(exactly-one-match, all-distinct-exists), and the n=14 trace figure
into a new companion paper at
papers/dual_decomposition_iterated_reduction/. Figures and figure
scripts moved via git mv (history preserved).
- In the main paper, Section 3 ("An iterated reduction") becomes
Section 3 "Cubic-graph edge contraction" (just the contraction
definition + 4-face theorem).
- Restructure Section 4 to host both the original face-monochromatic-pair
conjecture (clauses 1-3) and its strengthening (adds clause 4) as
separate conjectures, after briefly experimenting with folding them
into one. The empirical evidence is asymmetric (n<=21 for (1)-(3),
n<=18 for the full set), which the two-conjecture split presents more
honestly. The companion-paper reference is now in Section 4's intro.
Co-Authored-By: Claude Opus 4.7 <noreply@anthropic.com>
- Cut Conjecture 3.8 + Remark 3.9 from Section 3 and move into a new
Section 4 "The Four Colour Theorem from a strengthened conjecture".
- Add Remark 4.X spelling out the implication: clause (4)(i) forces the
cyclic colour pattern (c,a,c,b) on the new 4-face f_n, two opposite
edges of which satisfy the hypothesis of Theorem 3.9 verbatim; case
(ii) is conjecturally reducible to case (i) via a Kempe swap on the
{b,c}-cycle through X_1 X_2. Theorem 3.9 then produces the proper
3-edge-colouring of the contraction, contradicting minimality of G.
- Rewrite the bridge prose into the cubic-contraction definition to
reference Section 4 forward, rather than the conjecture directly.
Co-Authored-By: Claude Opus 4.7 <noreply@anthropic.com>
- Conjecture 3.6: add the 4-edge-face criterion as clause (3), with empirical
table through n=21 (complete, 535,182/535,182 pass) plus partial n=22
(641,700 colourings, timed out).
- Conjecture 3.8: strengthening with clause (4) on the b,c-Kempe cycle / 3-colour
alternative on the new face f_n; existential at the witness level. Tested
through n=18 (13,800/13,800 pass).
- Definition + figure for cubic-graph edge contraction (delete edge, smooth the
resulting degree-2 endpoints; equivalent to simple contraction in the dual).
- Theorem: cubic contraction across a 4-face preserves 3-edge-colourability when
the two opposite boundary edges have different colours. Constructive proof:
the two smoothed-in edges inherit the colour of the w_i pair they absorb, and
e_1 is recoloured to the third colour.
- Add 2-panel illustration of the theorem's recolouring.
- Trim Remark 3.7 and 3.9 tables to fit within \textwidth.
Co-Authored-By: Claude Opus 4.7 <noreply@anthropic.com>
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>
- Update def:edge-names to distinguish side-0 ({A_i, v_n}) and side-1
({A_{i+2}, v_n}); merged and spike unchanged.
- Add a paragraph defining the {a,b}-Kempe cycle in a 3-edge-coloured cubic
graph.
- Add lem:kempe-spike: in any proper 3-edge-colouring of the reduced dual,
the {c, c_0}-Kempe cycle through the spike contains side-0 and merged
(symmetrically for side-1 with c_1).
- Proof by Kempe swap: a hypothetical alternative cycle K containing merged
but not spike would, after swapping c <-> c_0 on K, give a proper
3-edge-colouring under which spike and merged disagree --- contradicting
lem:chord-apex.
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>
- Name the edges of the reduced-dual construction (merged, spike, sides)
via a new definition; use these names in lem:chord-apex.
- Add lem:pentagonal-externals with full exhaustive proof: any proper
3-edge-colouring near a pentagonal face of a cubic plane graph has its
five external edges forming, up to cyclic rotation, the pattern
(a, b, c, c, c) with {a, b, c} = {1, 2, 3} (iff).
- Cite the new lemma in the chord-apex proof scaffold as the lifting step.
- Remove the icosahedron experimental remark.
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>
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