For each chord-apex+Kempe colouring, record:
- |V(K_b)|, |V(K_c)|, |V(K_b) cap V(K_c)|, |V(K_b) cup V(K_c)|
- "Flip count" on each cycle: #consecutive pairs whose third-colour
edges lie on opposite local sides (= #same-Heawood pairs by Lemma A).
Results (n in [12, 18], 13,800 colourings):
- |V(K_b) cap V(K_c)| is NEVER 2 -- always >= 6. The two Kempe cycles
through merged share many vertices.
- Distributions of flip_Kb and flip_Kc are identical multisets
(consistent with b <-> c symmetry of the construction).
- But per-colouring, flip_Kb == flip_Kc only 39.65% of the time --
the symmetry is statistical, not pointwise.
- Max observed flip count is 20, never the maximum possible |V(K)|.
Consistent with h_phi never being constant on V(K_b) U V(K_c).
The substantial overlap of K_b and K_c (>= 6 shared vertices) means
the constancy hypothesis would impose simultaneous alternation
constraints from both cycles at every shared vertex -- the topological
"trap" the proof needs to exploit.
Co-Authored-By: Claude Opus 4.7 <noreply@anthropic.com>
Two diagnostic scripts probing the side-classification of c-edges at
K_b-vertices and their relationship to Heawood numbers:
1. check_heawood_side_correlation.py (first attempt)
- Defines "side" as connected component of H \ K_b.
- Result: K_b separates H into 2 components in 0% of cases, so
this notion doesn't capture the planar side. (Negative result --
kept for the record / so we don't redo it.)
2. check_heawood_local_side.py (correct version)
- Defines "side" locally via the planar CW embedding at v: c-edge
is on local RIGHT if, going CW from incoming K-neighbour at v,
we hit the c-neighbour before the outgoing K-neighbour; local
LEFT otherwise.
- Result on 625,200 consecutive K_b-pairs across 13,800
chord-apex+Kempe colourings (n in [12, 18]):
same h, same side: 0
same h, diff side: 372,456 (59.57%)
diff h, same side: 252,744 (40.43%)
diff h, diff side: 0
The empirical biconditional holds perfectly:
h_phi(v_0) == h_phi(v_1) <==> c-edges on opposite sides
This is "Lemma A" -- the corrected version of the proposed
orientation lemma. Equivalently: constant Heawood on a Kempe
cycle K forces the c-edges (off-K) to ALTERNATE inside/outside
of K along the cycle (not all on one side as I initially
conjectured).
This empirical result revises the spiral picture for Path 4: under
the Lemma 5.3 hypothesis of constant h on V(K_b) U V(K_c), the
c-edges alternate sides on K_b (and the b-edges alternate sides on
K_c). K_c must then cross K_b at every K_b-vertex it shares -- a
strong topological constraint we can now exploit.
Co-Authored-By: Claude Opus 4.7 <noreply@anthropic.com>
constancy on V(K_b) U V(K_c))
Three empirical checks on all chord-apex+Kempe colourings up to
n = 20 (142,812 colourings):
1. check_heawood_on_kempe.py
- Sum_v h_phi(v): not zero in general; 17.6% of colourings have
sum 0, the rest range in {+-4, +-8, +-12, +-16, +-20, +-24}.
So the global "Heawood sum = 0" identity fails.
- h_phi constant on V(K_b) U V(K_c): NEVER (0/142,812). This is
the central empirical result -- by Lemma 5.3's contrapositive
it gives an empirical proof of Conjecture 5.1 on these
surrogates.
2. check_heawood_per_kempe_cycle.py
- Sum_{V(K_b)} h_phi and sum_{V(K_c)} h_phi range widely (-20 to
+20), with only ~23% zero. So the "Heawood sum on each Kempe
cycle = 0" identity also fails -- the per-cycle sum is not the
right invariant.
3. check_heawood_pair_mismatch.py
- For each of 16 named-vertex pairs (v_n with each A_j, A_j with
A_k for j, k in {i, ..., i+4}), counts how often h_phi differs.
No pair is *always* differing -- the closest are consecutive
pairs (A_j, A_{j+1}) at ~75% diff. So the Heawood mismatch
enforcing non-constancy on V(K_b) U V(K_c) is diffuse, not at
a fixed pair.
Together these results confirm Path 4 (Conjecture 5.1 reduces via
Lemma 5.3 to showing h_phi non-constant on V(K_b) U V(K_c)) but
rule out the simplest single-pair-identity proof; the structural
obstruction lives elsewhere (likely a topological/cycle-winding
argument or a chord-apex/Kempe-spike colour cascade).
Co-Authored-By: Claude Opus 4.7 <noreply@anthropic.com>
Follow-up to Lemma 5.2. States that if Conjecture 5.1 has no
clauses-(1)-(3) witness for (G, G'^_{v,i}, phi), then h_phi is
constant on both Kempe cycles through merged, and the two constants
agree (since merged is on both cycles, so its endpoints force the
constants to match).
Proof is the V1-direction of the case analysis: differing h_phi on
either K_b or K_c reproduces a clause-(1)-(3) witness by the same
F_R/F_L geometry as Lemma 5.2's proof but with the hypothesis
"h_phi(v_0) != h_phi(v_1)", under which the matching-colour edges
land on the SAME face of e. Case B's merged-incidence corner is
handled by choosing a differing-Heawood pair away from merged's
endpoints.
Co-Authored-By: Claude Opus 4.7 <noreply@anthropic.com>
The previous statement "Heawood is constant on K through merged" was
strictly stronger than what the proof actually established without
Conjecture 5.3. Restate the lemma in the contrapositive direction:
If h_phi is constant on V(K), then no edge e in E(K) admits a face
F of G'^hat and edges e_1, e_2 on dF realising the clause-(3) arc
of Conjecture 5.1 at the endpoints of e.
Proof structure is mostly preserved (same F_R/F_L geometry, same case
split on phi(e) in {a, b}, same reading-off of cyclic colour orders).
The hypothesis "h_phi(v_0) != h_phi(v_1)" becomes "h_phi(v_0) =
h_phi(v_1)", which flips the conclusion: the same-coloured non-e
edges at v_0, v_1 land on opposite faces of e instead of the same
face. No dependency on Conjecture 5.3 or Theorem 4.X.
Redraw the figure to match the new lemma: both vertices labelled
h_phi = +1, both showing CW order (a, b, c), and the same-colour pair
(b-edges in Case A, a-edges in Case B) drawn on opposite sides of e.
Co-Authored-By: Claude Opus 4.7 <noreply@anthropic.com>
- Add Definition 3.1 "Heawood number of a vertex" (+1 if CW colour order
is (1,2,3), -1 if (1,3,2)) and cite Heawood 1898 in the bibliography.
- Add Lemma 5.2 "Heawood number is constant on the Kempe cycles through
the merged edge", positioned immediately after Conjecture 5.1. Its
proof exhibits a (F, e_1, e_2) witness for clauses (1)-(3) of the
conjecture from any pair (v_0, v_1) of consecutive K-vertices with
differing Heawood signs, by cases on whether phi(e) = a or b. The
proof does not invoke Conjecture 5.3 or Theorem 4.X.
- Add a two-panel figure illustrating Case A (b-edges on F_R when
phi(e) = a) and Case B (a-edges on F_L when phi(e) = b), with the
cyclic colour orders (a, b, c) at v_0 and (a, c, b) at v_1 visible
from the angular layout.
Co-Authored-By: Claude Opus 4.7 <noreply@anthropic.com>
- Main paper: dual_decomposition_minimal_counterexamples/ ->
face_monochromatic_pairs/. Title is now
"Face-Monochromatic Pairs and the Four Colour Theorem".
- Companion paper: dual_decomposition_iterated_reduction/ ->
iterated_reduction_in_reduced_dual/. Title is now
"An Iterated Reduction in the Reduced Dual". Its prose and bibliography
cite the parent under the new title.
- Update one absolute sys.path reference inside
check_conj_face_kempe_n15.py that pointed at the old folder.
Co-Authored-By: Claude Opus 4.7 <noreply@anthropic.com>
- 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>
Enumerate non-Hamiltonian cyclically-5-connected cubic planar graphs by
running plantri -c5 -d for n in {23,25,26} (n=24 already in the previous
commit) and filtering for non-Hamiltonian dual:
n=23 -> 0 of 1970 (recomputes Faulkner-Younger minimality)
n=24 -> 1 of 6833 (the Tutte/Fig 2.10 graph)
n=25 -> 1 of 23384 (new; unique 46-vertex one)
n=26 -> 0 of 82625
Both T (n=24) and T_25 (n=25) verified internally 6-connected by exhaustive
5-cut scan: every 5-cut is the neighborhood of a degree-5 vertex. This is
the strongest connectivity a planar triangulation can have and the level
at which Birkhoff-style reductions terminate, so both are genuinely
irreducible bases of any decomposition argument.
T_25 is also bridge-derived: witness Even Level Graph from source 24
(max level 4) at depth 2, orbit only 3114 states. Forward switches:
remove {21,23} add {22,24}; remove {3,5} add {1,6}. Both adds are bridges
of the even parity subgraph. Same witness signature as T (minimum total
Betti, tiny orbit, depth 2).
New subsection "Beyond n=24: enumeration and the next 5-connected core",
abstract extended, new Figure 7 (core_n25_dual.png). Reproducibility
scripts: draw_core_witness.py and verify_core_witness.py (both
parametrized so they work on any 5-conn non-Ham-dual core's g6).
Co-Authored-By: Claude Opus 4.7 (1M context) <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>
Record the partition sweep on the n=24 Fig 2.10 dual. New subsection +
experiments/bridge_partition_sweep.py.
Findings:
- A bridge switch is a constrained diagonal flip; bridge-derived via L
means lying in an Even-Level-Graph component of the restricted flip
graph. So the question is which flip-components contain an ELG.
- Identity: every 4-coloring of a triangulation has e_cross = 2n-4 (each
face has one within-pair edge), so total parity-subgraph Betti =
(c_A+c_B)-2; intertwining trees are the Betti-0 case.
- Of T's 333 valid partitions, total Betti splits 288/42/3 over 1/2/3;
min is 1 (T not intertwining). All 27 partitions found bridge-derived
(depth 2-3) have the minimum Betti 1 -> necessary.
- But not sufficient: only 27 of 288 Betti-1 partitions yield a witness;
the rest have flip-orbits >1.5e5 with no ELG, and a 12x budget increase
found none. The discriminator is flip-component structure (sharp
orbit-size dichotomy), not a numerical invariant. Characterizing which
Betti-minimal partitions sit in an ELG component is left open.
Co-Authored-By: Claude Opus 4.7 (1M context) <noreply@anthropic.com>
New paper introducing the dual (inner/weak dual) of a maximal planar graph,
dual depth (BFS-derived min level over a face's vertices), and a Tait-based
framing of a minimal 4CT counterexample via nested level duals. Includes a
dual-depth figure and its generator. Shelved per closing note in favour of an
alternative approach.
Co-Authored-By: Claude Opus 4.7 (1M context) <noreply@anthropic.com>
Retitle to "Even Level Graph Generators: a constructive conjecture
stronger than the Four Color Theorem" and state explicitly in the
abstract and introduction that the conjecture implies the four color
theorem but is strictly stronger: a 4-coloring grouped {1,2}|{3,4} is
exactly a partition into two bipartite-inducing parts, so 4CT is the bare
existence of such a partition, whereas the conjecture demands it be
realized constructively (bridge-switch level parity, or two induced
trees). Hence a proof is a new constructive proof of 4CT, and the
conjecture is at least as hard -- very likely harder.
Co-Authored-By: Claude Opus 4.7 (1M context) <noreply@anthropic.com>
Add the n=24 result to the Even Level Graph Generators paper: the dual of
the unique 44-vertex non-Hamiltonian cyclically-5-connected cubic planar
graph (Holton-McKay Fig. 2.10) -- a 24-vertex 5-connected triangulation,
the first conjecture test outside the 3-cut family -- is a bridge-derived
level graph, two verified bridge switches from an Even Level Graph
(source 19).
- Generate the graph rather than transcribe it: plantri -c5 lists all 6833
5-connected 24-vertex triangulations; exactly one has a non-Hamiltonian
dual, which also settles the uniqueness Holton-McKay left open at 44
vertices (cyclically-5-connected triangulation <=> dual cubic graph).
- New abstract sentence + "cyclically-5-connected case: n=24" subsection,
noting the classic 46-vertex Tutte graph is only cyclically 3-connected.
- Figure 6 (figures/fig210_dual.png): the dual T, parity-coloured, with the
two introduced bridge edges {6,19} and {20,22} in green (style of Fig. 5).
- Experiments: test_fig210_dual_bridge.py (generate->filter->test pipeline),
verify_fig210_witness.py (step-verifies the witness), draw_fig210_dual.py
(figure), fig210_dual.g6 (the unique graph). paper.pdf rebuilt (10 pages).
Co-Authored-By: Claude Opus 4.7 (1M context) <noreply@anthropic.com>
Records, for 4<=n<=11, triangulation iso classes, how many admit an ELG
source, ELG iso classes, and the automorphism-free flag-rooted count
sum_G 4E/|Aut(G)| * s(G). Computed by experiments/count_elgs.py.
Co-Authored-By: Claude Opus 4.7 (1M context) <noreply@anthropic.com>
count_elgs.py enumerates triangulation iso-classes and counts Even Level
Graphs (G,v) per n: iso-classes (sources up to Aut) and flag-rooted
(4E/|Aut| * s, an exact integer since Aut acts freely on flags).
Co-Authored-By: Claude Opus 4.7 (1M context) <noreply@anthropic.com>
experiments/test_tutte_bridge.py: bridge-derivability test for the dual of
the 46-vertex Tutte graph (a 25-vertex non-intertwining triangulation,
since the Tutte graph is non-Hamiltonian) -- the conjecture's first case
beyond the n=21 Holton-McKay duals. Reuses the fast integer-state bridge
engine: per source labelling with bipartite parity subgraphs, run a
backward bridge-orbit BFS for an Even Level Graph witness.
Co-Authored-By: Claude Opus 4.7 <noreply@anthropic.com>
The smallest non-Hamiltonian 3-connected cubic planar graphs have 38
vertices (J. Combin. Theory Ser. B 45:305-319). Reference for the
even_level_graph_generators conjecture: via the intertwining-tree =
Hamiltonian-dual equivalence, the duals of the non-Hamiltonian C3CPs
classified here are exactly the triangulations the conjecture must
reach through the bridge-derived-level-graph disjunct.
Co-Authored-By: Claude Opus 4.7 (1M context) <noreply@anthropic.com>
Frame the paper's purpose: ask whether two constructive families of
4-colorable triangulations -- bridge-derived level graphs (parity
2-coloring) and intertwining trees (two trees, disjoint color pairs) --
suffice to generate every maximal planar graph on n vertices. An
affirmative answer would be a constructive proof of the four color theorem
for triangulations. State the duality bridge to Tait/Holton-McKay and the
n=21 confirmation.
Co-Authored-By: Claude Opus 4.7 <noreply@anthropic.com>
Drop the n<=9 bridge-derived classification table and its surrounding
discussion; the n=21 boundary case now follows directly from the
trivial-below-21 observation.
Co-Authored-By: Claude Opus 4.7 <noreply@anthropic.com>
Remove the witness-ELG figure (former Fig. 5); keep the six resulting duals
with their introduced green bridge edges. Fix the dangling cross-reference
in the caption.
Co-Authored-By: Claude Opus 4.7 <noreply@anthropic.com>
Replace the radial (crossing-heavy) figure with two crossing-free planar
drawings (networkx planar_layout / Chrobak-Payne):
fig:n21-elgs -- the six witness Even Level Graphs, parity-coloured, with
the bridge-switch-flipped edges dashed red;
fig:n21-duals -- the six resulting duals, with the introduced bridge edges
solid green.
ELG and dual are drawn with independent planar layouts so neither has any
edge crossing (a flip diagonal would otherwise cross other edges when its
quadrilateral is non-convex, which happens for duals 0 and 3). Drop forced
equal aspect so panels fill and labels separate.
Co-Authored-By: Claude Opus 4.7 <noreply@anthropic.com>
Figure fig:n21-witnesses: each of the six Holton-McKay duals drawn as its
witness Even Level Graph in a radial-by-level layout (source centre,
level-k vertices on ring k), coloured by parity. Dashed red edges are the
flipped same-parity edges and solid green edges the introduced bridges;
applying the switches yields the dual. Duals 1,2 are ELGs outright.
draw_witnesses.py generates the combined 2x3 figure and per-dual PNGs from
the verified witness JSONs.
Co-Authored-By: Claude Opus 4.7 <noreply@anthropic.com>
Table tab:n21 records, for each of the six duals: not an intertwining tree;
Even Level Graph source (duals 1,2 only); and bridge-switch path length to
an ELG (0,0 for the two ELG-outright cases; 3,1,2,4 for the rest). All six
are bridge-derived; all witnesses step-verified.
Co-Authored-By: Claude Opus 4.7 <noreply@anthropic.com>
Tested duals 1 and 2: both are Even Level Graphs directly (dual 1 for
source 10, dual 2 for source 9), so bridge-derived with a zero-length
switch sequence. All six Holton-McKay duals are confirmed non-intertwining
(consistent with the dual-Hamiltonian theorem, since all six HM graphs are
non-Hamiltonian) and all six are bridge-derived. Saved witness files
dual_1.json, dual_2.json (0 switches) to complete the archive for all six.
Updated the n=21 subsection accordingly.
Co-Authored-By: Claude Opus 4.7 <noreply@anthropic.com>
- Conjecture now reads "bridge-derived level graph ... an intertwining tree,
or both" -- the stronger form the evidence actually supports (a bridge-
derived level graph is automatically a valid derived level graph).
- Empirical table recomputed for bridge-derivability, exhaustively for n<=9
(every backward bridge-orbit fully enumerable there):
n=7: 1 inter-only; n=8: 2 inter-only; n=9: 14 inter-only; missing=0.
Added prose: below n=21 every class is intertwining, so the table shows
how far the bridge-derived disjunct reaches on its own (36/50 at n=9) and
that the two disjuncts complement each other; "bridge only" is 0 in range.
- n=21 subsection notes the four witnesses are explicit, short (path lengths
3,1,2,4), archived, and step-verified.
Co-Authored-By: Claude Opus 4.7 <noreply@anthropic.com>
The hunt only logged partition indices; the actual witness ELGs were lost.
Re-extract them (deterministic) with full bridge-switch paths and verify
every step independently. Saved as experiments/witnesses/dual_<i>.json
(labels, ELG source, ELG + dual graph6 and edge lists, the explicit
remove/add bridge-switch sequence, verified flag). All four verify:
dual 0: ELG source 18, 3 bridge switches to dual
dual 3: ELG source 16, 1 bridge switch to dual
dual 4: ELG source 20, 2 bridge switches to dual
dual 5: ELG source 1, 4 bridge switches to dual
So each dual is only a handful of bridge switches from an Even Level Graph,
and the witnesses are now reproducible and human-checkable.
Co-Authored-By: Claude Opus 4.7 <noreply@anthropic.com>
Add the figures for the n=7, idx=2 missing-isomorphism case
(missing_iso_n7_idx2.png is included in paper.tex), plus its
4-coloring and level-decomposition companions and the G-for-T
preimage graph. Rebuild paper.pdf and its LaTeX aux/log/out.
Co-Authored-By: Claude Opus 4.7 (1M context) <noreply@anthropic.com>
Backward bridge-switch search (sharded over valid parity partitions) found
an Even Level Graph witness for each of the four previously-open duals:
dual 0: partition 12, witness orbit 9458
dual 3: partition 9, witness orbit 388
dual 4: partition 23, witness orbit 3842
dual 5: partition 12, witness orbit 165668
So all four are bridge-derived level graphs, hence valid derived level
graphs. Combined with the two duals that are Even Level Graphs outright,
the disjunction is now confirmed for ALL SIX critical iso classes at n=21
-- the first nontrivial test of the conjecture passes.
Why it worked where exhaustion failed: a witness, when it exists, tends to
sit in a SMALL orbit (here a few hundred to ~1.7e5 states) reachable
quickly, while other parity partitions of the same triangulation have
orbits >1e6. We only need one good partition. The bridge restriction both
shrinks orbits ~100x and guarantees validity, so any ELG found in a
backward orbit is an immediate witness.
- Update paper n=21 subsection to report the resolution.
- Add shard_hunt.py (partition-sharded parallel witness hunt).
Co-Authored-By: Claude Opus 4.7 <noreply@anthropic.com>
Relocate the standalone Python scripts from the repo root into the
experiments/ folder of the paper each one belongs to:
plane_depth_sequencing/experiments/
plane_depth_sequencing.py, draw_quad_sequence.py,
draw_quad_sequence_diagram.py, extract_sequence.py,
plane_depth_sequencing_figure.py, quad_sequence_coloring_check.py
colored_edge_flip_classes/experiments/ colored_edge_flip_class_survey.py
colored_pentagon_contractions/experiments/ colored_pentagon_contractions.py
plane_diamond_coloring/experiments/ plane_diamond_coloring.py
Each file that imports lib.* (still in the repo root) or the sibling
plane_depth_sequencing module gets a sys.path shim that prepends the
repo root (computed three levels up) and, where needed, its own dir.
Imports verified to resolve from a neutral working directory.
flip_symmetric_census.py is intentionally left in the root.
Co-Authored-By: Claude Opus 4.7 (1M context) <noreply@anthropic.com>
Findings at n=9 (50 triangulations, orbits fully exhaustible):
- 36 bridge-derived, 14 NOT bridge-derived. So bridge-derived is a PROPER
subclass of derived (49 derived at n=9). All 14 non-bridge graphs are
intertwining trees -- as are all 50, necessarily: intertwining tree
<=> dual Hamiltonian, and the smallest non-Hamiltonian 3-connected cubic
planar graph has 38 vertices, i.e. dual on 2n-4=38 => n=21. Hence every
triangulation with n<=20 is an intertwining tree, and the disjunction
"bridge-derived OR intertwining" is trivially true below n=21. The 4
Holton-McKay duals are the first non-intertwining triangulations.
- Static parity-subgraph invariants (Betti numbers, component counts,
cross-edge count, existence of an all-forest partition) do NOT separate
bridge-derived from non-bridge-derived -- both classes realize beta=0
partitions and identical ranges. Bridge-derivability is dynamical, not a
simple static invariant; no easy obstruction.
- Side lemma: every valid parity partition of an n-vertex triangulation has
exactly 2n-4 cross edges (intra-edges = n-2). Holds for all n=9 graphs.
Co-Authored-By: Claude Opus 4.7 <noreply@anthropic.com>
- fast_bridge.py: states as 210-bit integer edge-bitmasks (compact memory,
O(1) set ops); build a NetworkX graph only once per state for the planar
embedding; parity-subgraph bridges via one iterative DFS per state instead
of per-edge subgraph copies. Validated identical orbits to the slow version;
throughput ~5170 states/s vs ~1100 (graph.copy was 66% of old runtime).
- fast_decide.py: integrated, gated ELG-witness check (only even-class
sources with all-opposite-class neighbourhoods are tested with the
ground-truth is_even_level_graph, then parity match). Witness detection
validated (ELGs -> True, T*_9 -> False).
- Feasibility finding: bridge orbits are ~100x smaller than full E/O orbits
but still 1e5-1e6 states per labelling (partitions 0,1 of dual 0 exceed
310k and 685k without exhausting), x ~150 valid parity partitions per dual.
Exhausting every orbit -- required for a conclusive NEGATIVE -- is
computationally infeasible. A conclusive POSITIVE (witness ELG) remains
reachable; none found so far.
Co-Authored-By: Claude Opus 4.7 <noreply@anthropic.com>
- Update abstract and coverage table: computational evidence now includes n=12
(previously n=6..11). All iso-classes remain reachable.
- Correct conjecture statement: minimum degree ≥5 (not ≥4).
- Add graphicx package (for potential figure support).
- Add exploratory experiment files for exception characterization, preimage
search, and visualization (directed toward understanding the full orbit
of the T*_9 case and related structural questions).
Co-Authored-By: Claude Opus 4.7 <noreply@anthropic.com>
- Define bridge switch (E/O switch whose new same-parity edge is a bridge
in its parity subgraph) and bridge-derived level graph in the paper.
Note that bridge switches preserve bipartite parity subgraphs, so every
bridge-derived level graph is automatically valid.
- Discover the E/O-switch relation is directed (irreversible when a switch
produces a cross-parity edge); T*_9 reaches an ELG forward but no ELG
reaches it, explaining why it is not derived. This rules out a simple
switch-invariant characterization.
- Bridge orbits are far smaller than full E/O orbits (~10^4 vs ~10^8 for
some labellings), making exhaustive search feasible. Each of the 4 open
duals has ~150 valid parity partitions; exhaustive bridge-orbit search
per partition can decide bridge-derivability conclusively.
Co-Authored-By: Claude Opus 4.7 <noreply@anthropic.com>
- Add Theorem: maximal planar G is an intertwining tree iff its dual
G* is Hamiltonian (Tait-style Jordan-curve argument). Consequence:
smallest non-intertwining-tree triangulations are the 6 duals of the
38-vertex Holton-McKay graphs, at n=21.
- Load the 6 graphs from McKay's authoritative planar_code file
(nonham38m4.pc), verified: 38 vertices, cubic, planar, non-Hamiltonian.
- All 6 duals confirmed not intertwining trees (exhaustive 2^20 check).
- 2 of 6 duals are themselves Even Level Graphs (sources 9, 10), hence
derived level graphs -- first cases where the derived disjunct does
work the intertwining-tree disjunct cannot.
- Remaining 4: bounded E/O-orbit search inconclusive; status open. This
is the first genuinely undetermined instance of the conjecture.
Co-Authored-By: Claude Opus 4.7 <noreply@anthropic.com>
Disjunction (every maximal planar graph is a derived level graph or
intertwining tree) holds through n=12. New intertwining-only iso class
at n=12 (analog of T*_9 at n=9) brings the count of derived-resistant
iso classes to 2 in this range. Per the intertwining-tree ⟺
Hamiltonian-dual equivalence, intertwining-tree failures cannot occur
until n=21 (dual of the 38-vertex Holton-McKay minimum Tait
counterexample).
Co-Authored-By: Claude Opus 4.7 <noreply@anthropic.com>
didericis