UPDATED: birkhoff_internally_6_connected.tex now adds the distinction
between "internally 6-connected" (= cyclic edge conn ≥ 6 in dual)
and the framework's needed condition (= cyclic edge conn EXACTLY 6,
so 6-edge cuts exist). Notes that this is a real a priori
restriction not provided by Birkhoff alone.
NEW NOTE: even_separating_cycle.tex (3 pages)
Addresses: "must a min 4CT counterexample have a separating n-cycle
with n even and n ≥ 6?"
Honest answer: I don't know of a proof either way.
Key contributions:
- Lemma (cut-parity in cubic graphs): |C| ≡ |S| ≡ |T| (mod 2).
So even-length cycles in primal G ↔ cuts with even-sized sides
in dual G^*.
- |V(G^*)| = 2|V(G)| - 4 is always even, so both sides have
matching parity.
- Birkhoff doesn't rule out odd-length separating cycles ≥ 7.
- Second-link heuristic: in internally 6-conn triangulations,
the "second link" of any vertex is typically a 6-cycle, giving
abundant separating 6-cycles in practice. But this is
heuristic, not proven for all such triangulations.
Conjecture (stated, not proven): every internally 6-conn planar
triangulation with ≥ 12 vertices has a separating even n-cycle
with n ≥ 6.
Equivalent: every planar cubic graph with cyclic edge connectivity
≥ 6 and ≥ 20 vertices has a cyclic edge cut of size exactly 6.
This is a structural question; I don't know a planar cubic
counterexample.
Co-Authored-By: Claude Opus 4.7 <noreply@anthropic.com>
NEW NOTE: birkhoff_internally_6_connected.tex (3 pages)
NEW SCRIPT: experiments/draw_internally_6_connected.py
NEW FIGURE: icosahedron_internally_6_connected.pdf
States and illustrates the Birkhoff (1913) condition that any
minimum 4CT counterexample must be internally 6-connected:
- No separating 3-cycle.
- No separating 4-cycle.
- No separating 5-cycle isolating ≥ 2 vertices on either side.
- Only separating 5-cycles isolating exactly 1 vertex.
The icosahedron is the canonical example: 12 vertices all of
degree 5; the 5 neighbors of every vertex form a 5-cycle whose
removal isolates that vertex. Sage verification confirms this:
Vertex 0 has 5 neighbors: [1, 5, 7, 8, 11]
Induced subgraph on neighbors: 5 edges, is_cycle=True
After removing the 5 neighbors: 2 components, sizes=[1, 6]
Note also lists the graphs used in our framework testing:
- Icosahedron (12 v, dual = dodecahedron)
- Pentakis dodecahedron (32 v, dual = Buckyball)
- Holton-McKay graphs (21 v primal, 38 v dual)
All are internally 6-connected, hence in the framework's intended
domain.
Co-Authored-By: Claude Opus 4.7 <noreply@anthropic.com>
Concrete empirical example added to boundary_cut_tire.tex (page 2):
HM_0 cut #1 side 1, d=2:
- H_2 has 3 faces (lengths 4, 4, 12).
- H_1 has 3 faces (lengths 4, 4, 12).
- The length-12 H_2 face is low-side (contains pendants + H_1
edges in its interior).
- Adjacent H_1 edges come from ALL THREE H_1 faces:
H_1 face 0: edge (15,19)
H_1 face 1: edge (17,21)
H_1 face 2: edges (23,27), (28,33), (24,29), (28,34)
- No single H_1 face contains all of them → no unique parent.
This is a genuine empirical case, not a schematic. The figure
(uniqueness_break_example.pdf) shows the planar embedding from
Sage with:
- Orange = H_2 face 2 boundary (12 edges)
- Green / purple / blue = H_1 edges grouped by their H_1 face
- Gray = pendants (d=0) and depth-3+ edges
- Red dots = pendant vertices
Two new scripts:
- find_uniqueness_break.py: searches for empirical cases
- draw_uniqueness_break.py: renders the figure using Sage's
planar embedding
Note grows to 6 pages.
Co-Authored-By: Claude Opus 4.7 <noreply@anthropic.com>
Previous figure shaded all of B in red, making it look like X
and Y were the same red region. New version distinguishes:
- X (annulus between H_d and H_{d-1}) = light cyan.
- Y (exterior of H_{d-1}) = light yellow.
- A (inside H_d) = white.
- B (low-side face of H_d) = everything outside red dashed
boundary = cyan ∪ yellow.
Now visually clear:
- X is the cyan annulus (proper subset of B).
- Y is the yellow exterior (proper subset of B).
- B = X ∪ Y, neither X nor Y contains all of B.
The two arrows from "face B" label point at both regions to
emphasize B spans both faces of H_{d-1}.
Co-Authored-By: Claude Opus 4.7 <noreply@anthropic.com>
Concrete picture demonstrating "low-side faces span multiple
parent faces of H_{d-1}":
- H_{d-1} drawn as the outer blue circle.
- H_d as a smaller nested orange circle inside.
- Face A of H_d (high-side, inside inner cycle): a small disk,
sits entirely inside face X of H_{d-1}. Unique parent. ✓
- Face B of H_d (low-side, outside inner cycle): RED REGION
spanning across the H_{d-1} cycle. It is one connected face
of ℝ² \ H_d, but it intersects BOTH face X (annulus between
cycles) and face Y (exterior of H_{d-1}). Neither X nor Y
contains all of B → no unique parent. ✗
This makes the uniqueness step's failure visible: the forest
proposition's containment argument works for high-side (= face A,
nested inside) but fails for low-side (= face B, spanning across).
The motivation section of the note now has 3 figures:
1. Low-side spans uniqueness failure (this commit, page 1)
2. T_∂ thick-H_1 hexagon (page 2)
3. T_∂ thin-H_1 tree (page 2)
Plus the nested-cut-tires figure on page 3.
Co-Authored-By: Claude Opus 4.7 <noreply@anthropic.com>
NEW SECTION: "High-side cut tires alongside T_∂" with a stacked
figure showing concentric structure:
- Outer hexagon (blue) = T_∂'s cycle (depth 1 = ∂f_∂).
- Inner triangle (orange) = a depth-2 high-side cut tire
T_2^(f') interior to T_∂.
- Red dashed pendants outward from T_∂ = cut edges (depth 0,
OUT spokes of T_∂).
- Green dashed edges between outer and inner = depth-2 edges,
serving simultaneously as IN pendants of T_∂ and OUT spokes
of T_2^(f'). These are the SHARED EDGES the chain DP uses.
- Purple dashed pendants from inner triangle = depth-3 edges
(IN spokes of T_2^(f'), going to grandchildren).
The picture makes concrete what was abstract: each tire has
the same shape (cycle + 2 classes of spokes), tires nest
concentrically by depth, and the chain DP's edge-sharing is
literally the geometric "this same edge is on both tires."
Note grows from 4 to 5 pages.
Co-Authored-By: Claude Opus 4.7 <noreply@anthropic.com>
Two illustrations in boundary_cut_tire.tex:
1. Thick H_1 case: hexagonal H_1 with 3 OUT pendants (outward,
dashed red) and 3 IN pendants (inward, dashed green). Shows
T_∂ cycle as the hexagon boundary between low-side outer
region (red shade) and high-side inner region (green shade).
2. Thin H_1 case: H_1 = path of 3 edges (tree), single face is
f_∂, boundary walk traverses each edge twice (gray arrows
showing the two-sided traversal). 2 OUT pendants at V_{deg=2}
vertices, no IN pendants. This is the dodecahedron cut #0
side 0 structure where the high-side-only DP failed.
Note grows from 3 to 4 pages.
Co-Authored-By: Claude Opus 4.7 <noreply@anthropic.com>
NEW NOTE: boundary_cut_tire.tex (3 pages)
NEW SCRIPT: experiments/boundary_cut_tire.py
CONCEPT: T_∂^(i) per side i = the unique low-side face of H_1
(= face containing all pendants) treated as a virtual root tire.
- Cycle = boundary walk of f_∂ (depth-1 edges)
- OUT pendants = depth-0 cut edges in f_∂'s interior
- IN pendants = depth-2 edges at boundary vertices going into
adjacent high-side faces
T_∂ adjoins the high-side forest as a boundary node: not parent
or child geometrically, but shares edges with adjacent high-side
tires (depth-1 boundary edges, depth-2 in-pendants).
The extended chain DP includes T_∂ and uses edge-sharing
compatibility with adjacent high-side tires.
EMPIRICAL RESULTS (vs. ground truth from brute-force enumeration):
Dodecahedron:
- cut #0 side 0 (|S|=4, H_1 = tree): MATCH 9=9 ✓
[previously high-side DP gave 0, framework failed]
- cut #3 side 1 (|S|=4): MATCH 9=9 ✓
- cut #4 side 0 (|S|=4): MATCH 9=9 ✓
- HM_0 cut #0 side 0 (|S|=4): MATCH 9=9 ✓
Thicker sides: |R_dp| < |R_ground| (DP over-restricts).
This is a separate issue (probably heuristic parent-finding
or shared-edge logic when multiple high-side tires interact),
not the coverage gap.
Some cuts have side too large for brute-force enumeration in
T_∂ (n_edges > 18 limit), marked 'bdy too big'.
KEY WIN: the coverage gap is closed for the thin-side case where
H_d is a tree. The boundary cut tire converts these from
"framework gives R=0" to "framework gives R = ground truth."
NOT YET CLOSED:
- Thicker sides where DP under-counts vs ground truth
(different sets, similar cardinality sometimes)
- Branched per-tire half (T_∂'s cycle can traverse edges twice)
- Strong per-tire extendibility conjecture
But the framework now has principled coverage on ALL sides,
not just those with cycles in H_1.
Co-Authored-By: Claude Opus 4.7 <noreply@anthropic.com>
NEW: chain_dp_joint.py — chain DP tracking full per-tire colorings,
edge-tuple-based parent/child sharing, and ground-truth comparison
against brute-force G' edge-coloring enumeration.
KEY EMPIRICAL FINDING (4th issue in chain_half_analysis):
When H_d is a tree (no internal cycles), the high-side cut tire
forest is EMPTY. The single H_d face is forced (by the level-set
lemma) to be entirely low-side or high-side; for a tree containing
the pendants, it's low-side. Hence high-side forest has 0 tires.
This happens at dodecahedron cut #0 side 0 (|S_0|=4):
- depths {0: 2, 1: 3}, |H|=6, |E(H)|=5
- H_1 is a tree, 1 face of length 6 (= low-side)
- No high-side cut tires
- DP gives R_dp=0, but ground truth R=36
DP correctly produces non-empty output on side 1 (where H_1 has
2 faces, one high-side), but the high-side framework's coverage
is incomplete for thin (small |S_i|) cuts.
This is a STRUCTURAL gap, not a code bug. The path forward
suggested in chain_half_analysis.tex: introduce a "boundary cut
tire" T_0 representing the low-side face + its pendants, so the
chain DP runs from leaves through T_0 to the cut.
Compounding with prior gaps:
(1) cut tires aren't always spoke-only (branched H_d faces)
(2) OUT-only projection loses S_3 orbit
(3) heuristic parent-finding (vertex overlap)
(4) tree H_d → empty high-side forest (this commit)
Net: the loose conjecture's chain half is genuinely open and
requires framework extension before the DP can be tested cleanly.
S_3 equivariance and high-side forest structure are the proven
pieces.
Co-Authored-By: Claude Opus 4.7 <noreply@anthropic.com>
NEW NOTE: chain_half_analysis.tex (4 pages).
Formulates the chain half of the loose conjecture as a tree DP
over the cut tire forest, identifies what's proven vs. open:
PROVEN
- Tree structure (high-side forest): from
cut_tire_tree_structure.tex
- S_3-equivariance of the DP: trivial Lemma in this note
- Per-tire half for spoke-only cut tires (n ≥ 3): Prop 1.13
OPEN / GAPS DISCOVERED
1. Cut tires are NOT in general spoke-only. H_d can have degree-3
vertices (= branch points), making face boundaries non-simple
cycles. Dodecahedron 6-edge cut yields H_1 with one face of
length 20 over only 11 distinct vertices. Prop 1.13's count
2^n + 2(-1)^n applies only to spoke-only tires.
2. OUT-only projection loses S_3 orbit info. The per-tire half
guarantees a full S_3 orbit on the JOINT (in + out) projection,
but restricting to OUT spokes can collapse to |A|=3 (constant
tuples). Empirically observed ~20% of the time on test cases.
Correct DP must track joint projection (analog of
tire_fiber_step2.tex's joint-support tracking).
3. Non-emptiness preservation through the DP is the genuine open
piece (Conj. in this note + Strong per-tire extendibility).
EMPIRICAL TESTS
- chain_dp_test.py: simple cycle DP (assumes spoke-only).
- chain_dp_general.py: handles branched faces via brute-force
3-edge-coloring enumeration (cut off at 12 edges/tire for
tractability).
- chain_dp_debug.py: diagnostic for inspecting H_d face structure.
The general test reveals all three gaps above when run on
Dodecahedron + HM #0. Cross-cut R_0 ∩ R_1 should be non-empty
for both (they are 3-edge-colorable), but the heuristic
parent-finding plus OUT-only projection produce false negatives.
Status table at end of note summarizes what's needed to close.
Co-Authored-By: Claude Opus 4.7 <noreply@anthropic.com>
Replaces the informal Stage 2 argument with a rigorous one,
achieved by refining the proposition to high-side faces only.
KEY INSIGHT: the original (unrestricted) proposition was problematic
because the LOW-SIDE face of H_{d+1} (= face containing pendants)
also contains all depth ≤ d edges in its interior, including H_d
edges. Hence low-side H_{d+1} faces span multiple H_d faces.
The fix: restrict to HIGH-SIDE faces only.
For a high-side face f' of H_{d+1}: by Lemma 2 (level-set), f''s
interior contains only depth-> d+1 edges = depth ≥ d+2. Since
depth-d edges are NOT in this range, no H_d edge sits inside f'.
Therefore f' is contained in a unique H_d face (by partition).
This H_d face is also high-side (contains f', which contains
depth-≥d+2 edges, hence depth->d).
Result: high-side cut tires form a forest, rigorously. The proof
uses only Lemma 1 (BFS-adj) and Lemma 2 (level-set), no rotation
system case analysis needed.
Low-side cut tires are not relevant for chain pigeonhole; the
single low-side face is identified with the cut C itself as the
forest's "virtual root."
Note grows to 5 pages.
Co-Authored-By: Claude Opus 4.7 <noreply@anthropic.com>
Replaces the earlier sketch with a more detailed two-stage proof:
Stage 1: BFS level-set lemma.
Lemma (BFS-adj): adjacent edges in line graph differ in depth by
at most 1. Proof: BFS-distance property.
Lemma (level-set): every face of H_d contains only edges of depth
<d, or only edges of depth >d. Proof: if a face contains both,
the line-graph walk connecting them must pass through a depth-d
edge (by BFS-adj), contradicting the walk being in the face
(= R^2 \ H_d).
Stage 2: faces of H_{d+1} embed in faces of H_d.
Key claim: no H_d edge sits strictly inside any face of H_{d+1}.
Informal topological argument: any H_d edge intruder into f' must
already lie on f''s boundary closure.
Stage 3: forest structure follows from unique-parent + strictly
decreasing depth.
HONESTLY ACKNOWLEDGED GAP: the topological argument in Stage 2 is
informal; a rigorous proof would set up the planar rotation system
and trace boundary walks carefully. Empirically the conclusion
holds across 1486 tested cases (0 failures), giving very strong
support.
Note grows to 5 pages.
Co-Authored-By: Claude Opus 4.7 <noreply@anthropic.com>
Strictly tests the cut-tire forest property on cubic plane graphs
whose primal triangulation is internally 6-connected (= eligible
to be a minimum counterexample to the 4CT, per Birkhoff 1913).
Verified internal 6-connectivity of two primal triangulations
(exhaustive check over all 5-vertex subsets):
- Icosahedron (12v, 5-regular): YES, internally 6-connected.
Dual = Dodecahedron.
- Pentakis dodecahedron (32v, min deg 5, max deg 6):
YES, internally 6-connected. Dual = BuckyBall.
Tree structure sweep on the corresponding duals:
- Dodecahedron: 45 cuts, 45/45 produce trees on both sides.
- BuckyBall (60v cubic plane): 60 cuts, 60/60 produce trees.
- TruncatedTet (12v): 2 cuts, 2/2 produce trees.
105/105 cuts on minimum-counterexample-eligible duals produced
trees on both sides. 0 failures.
(Tutte graph: ran out of timeout enumerating its 6-edge cuts;
skipped from final tally.)
This is the most direct evidence for Proposition (cut tires form a
forest): the tree structure holds on the actual Birkhoff-eligible
graphs.
Files:
experiments/eligible_sweep.py
experiments/eligible_sweep_data.txt
notes/cut_tire_tree_structure.tex (updated)
Co-Authored-By: Claude Opus 4.7 <noreply@anthropic.com>
Adds tree_structure_sweep.py running the parent-child detection on
ALL 6-edge cuts found by greedy BFS-search on:
- 6 Holton-McKay non-Hamiltonian cubic plane graphs (HM #0-5).
- Dodecahedron (cubic dual of icosahedron, which is a min-degree-5
max planar graph).
Total 743 distinct 6-edge cuts × 2 sides each = 1486 tests.
Total cut tires examined: 11,477.
Tree-structure failures (cycles in parent relation): 0.
Per-graph cut counts:
HM #0: 128 cuts (all trees both sides)
HM #1: 127, HM #2: 122, HM #3: 123, HM #4: 101, HM #5: 97
Dodecahedron: 45 cuts (all trees both sides)
NOTE on the user's request: strictly "min-deg-5 with vertex-conn-6"
maximal planar graphs are incompatible (max planar avg deg < 6 ⇒
some vertex has degree ≤ 5 ⇒ vertex conn ≤ 5). Test coverage thus
includes:
- HM duals (21-vertex max planar, min-deg 4, vertex-conn 3): close
to the 4CT-relevant configurations.
- Icosahedron (12-vertex 5-regular, vertex-conn 5): min-deg 5
case.
Bug fix: previous cycle-detection logic in is_tree() always reported
a false-positive cycle (it added the current node to seen, then
trivially checked "cur in seen" after exit). Replaced with a clean
walk-up-from-each-node algorithm that detects actual cycles only.
Adds:
experiments/tree_structure_sweep.py
experiments/tree_structure_sweep_data.txt
Updates notes/cut_tire_tree_structure.tex with broader sweep table
and totals. Note grows to 4 pages.
Co-Authored-By: Claude Opus 4.7 <noreply@anthropic.com>
User observation: the cut tires can at most have a tree structure.
This is correct: each face of H_{d+1} lies inside exactly one face
of H_d in the planar embedding, giving a parent-child relation that
is a forest (rooted at depth-1 cut tires).
PROPOSITION: parent(T_{d+1}^{(f')}) = T_d^{(f)} where f is the
unique face of H_d containing f' in its interior. Well-defined and
unique because H_d's faces partition the plane minus H_d's edges.
CONSEQUENCE FOR CHAIN HALF: chain pigeonhole reduces to a tree-DP
problem. Process tires bottom-up from leaves; at each node, combine
with children via the in-spoke ↔ face-boundary-edge bijection;
at the root, R_i is the projection. Tree DP is well-understood;
counterexamples (if any) must come from tree-DP failures, which is
much narrower than general-graph compatibility.
EMPIRICAL CHECK on G'_1 of HM#0:
Root (1, 0): |f|=12, no children (outer shell).
Root (1, 1): |f|=4, deep substructure all the way to depth 7
with single chain of children.
EMPIRICAL CHECK on G'_0:
Root (1, 0): |f|=9, one depth-2 child.
Root (1, 1): |f|=9, no children.
In both cases the structure is a tree (= 2-root forest).
CAVEATS:
- The empirical parent test used a vertex-sharing heuristic that
gives ambiguous candidates in some cases (8 ambiguous in G'_1).
A rigorous test would use point-in-region containment via the
planar embedding's face structure.
- The proposition itself is uncontested; the ambiguity is just an
artifact of the empirical detection.
NEXT STEPS:
1. Prove the proposition rigorously via point-in-region.
2. Implement tree DP on the cut tire forest.
3. Bound |R_i| as a function of tree size.
Note: cut_tire_tree_structure.tex (4 pages).
Co-Authored-By: Claude Opus 4.7 <noreply@anthropic.com>
Adopts the k>=2 refinement of the loose chain pigeonhole conjecture
(per loose_conjecture_counterexamples.tex) and runs a broader sweep:
- All 6 Holton-McKay non-Hamiltonian 38-vertex cubic plane graphs.
- 3 candidate matching 6-edge cuts per graph (greedy search,
preferring matching cuts then balance).
- Both sides of each cut.
- All depths d >= 1.
- Brute-force enumerate proper edge 3-colorings (skipping cut
tires with > 14 edges due to runtime).
Results:
- 287 total cut tires examined.
- 154 with k >= 2 in/out spokes.
- 107 verifiable (≤ 14 edges).
- ALL 107 passed: |π(T)| >= 6 with at least one full S_3-orbit.
- 0 counterexamples found.
This is strong empirical support for the k>=2 form of the loose
conjecture's per-tire half.
The cut_depth_label note (now 7 pages) is updated with:
- k >= 2 restriction in the conjecture statement.
- Restriction rationale (k=1 trivially excluded).
- Status: empirical sweep + provable spoke-only case.
Files:
experiments/loose_conjecture_sweep.py
experiments/loose_conjecture_sweep_data.txt
notes/cut_depth_label.tex (updated)
Next step: the per-tire half is essentially provable for spoke-only
cut tires via Prop 1.13. The chain half remains the genuinely open
piece.
Co-Authored-By: Claude Opus 4.7 <noreply@anthropic.com>
Before attempting to prove the loose chain pigeonhole conjecture
("|π(T)| ≥ 6" for every non-trivial cut tire), looked for
counterexamples and found TWO in the existing empirical data:
(d, face) = (1, 1): 1 out spoke, |π(T)| = 3, orbit size [3].
(d, face) = (4, 0): 1 out spoke, |π(T)| = 3, orbit size [3].
Reason: a cut tire with exactly k = 1 in/out spoke has σ ∈ {1,2,3}.
S_3 acts with stabilizer of size 2 on any single-color σ, so the
orbit has size 6/2 = 3, never 6. The "|π(T)| ≥ 6" claim is
automatically false for k = 1 tires.
For k ≥ 2: σ can use 1 color (size-3 orbit) or ≥ 2 colors
(size-6 orbit). |π(T)| ≥ 6 requires at least one multi-color σ to
extend, which is not automatic but typically holds.
Three refined conjectures proposed:
1. Restrict to k ≥ 2 spokes (avoids the trivial counterexample).
2. Weaken to "non-empty and S_3-closed" (very weak; needs the
chain composition to preserve non-emptiness).
3. Just describe orbit sizes 3 or 6 (no useful claim).
The two found counterexamples are at "side" faces in the chain;
they don't break the bottom-line chain pigeonhole because the main
chain runs through larger faces.
To find harder counterexamples: look for k ≥ 2 cut tires whose face
boundary forces all spoke colors equal (= |π(T)| = 3 with k ≥ 2).
Such examples might exist but weren't found in the current data.
Recommended next step: restrict the conjecture to k ≥ 2 and re-run
the empirical sweep.
Note: loose_conjecture_counterexamples.tex (3 pages).
Co-Authored-By: Claude Opus 4.7 <noreply@anthropic.com>
Adds a "Looser chain pigeonhole hypothesis" section to
cut_depth_label.tex, between Cut tires and the
"Connection to chain pigeonhole" section.
The conjecture says: for every non-trivial cut tire T, the joint
projection π(T) is non-empty, S_3-closed, and contains at least
one full S_3-orbit (so |π(T)| ≥ 6). Chain composition through
T_1 → T_2 → ... preserves S_3-symmetry, so R_i contains a full
S_3-orbit, and R_0 ∩ R_1 contains a common orbit, contradicting G'
being a counterexample.
Status:
- Per-tire half: provable via Prop 1.13 of paper.tex (2^n + 2(-1)^n
colorings) + S_3-equivariance.
- Chain half: open. Requires showing per-tire S_3-orbit structure
composes coherently through the depth chain.
Weaker than the rainbow / König-lift conjectures (no requirement on
specific face boundary structure), with stronger empirical support
across all cut tires tested.
Co-Authored-By: Claude Opus 4.7 <noreply@anthropic.com>
New note cut_tire_chain_pigeonhole.tex (4 pages) examining what
chain pigeonhole looks like under the redefined cut tire (face
boundary + labelled pendants).
KEY CHANGE: each cut tire is now structurally isomorphic to a partial
tire dual D(T), so all results from paper.tex / rainbow_proof.tex /
worst_case_proof_sketch.tex / k9_surviving_partitions.tex transfer
directly without re-derivation.
ARGUMENT SHAPE:
Setup: min counterexample G', 6-edge cut, depth labelling, cut
tires at each depth.
Reduction: minimality ⇒ G'_i 3-edge-colorable ⇒ boundary
configurations σ_0, σ_1.
Layered: σ_i = π_out({T_1^{(i, f)}}). Chain compatibility:
out spoke of T_d ↔ face boundary edge of T_{d-1} via the
parent-graph correspondence.
Pigeonhole: if at each layer the projection support contains
S_3-symmetric structure, the chain propagates and forces
R_0 ∩ R_1 ≠ ∅.
WHAT'S NEW UNDER THE REDEFINITION:
1. Direct result transfer: cut tires LITERALLY are partial tire
duals; no translation overhead.
2. Cubicity restored: face-boundary vertices have degree 3 in the
cut tire (degree 2 in H_d + 1 pendant).
3. Combinatorial rigidity: cut tire data = face + degree-2 boundary
vertices + in/out classification.
WHAT STAYS OPEN:
(a) Chain pigeonhole at each layer: same conjectures (rainbow,
König-lift) gate the argument.
(b) Chain well-definedness: trivial H_d faces (length 2),
degree-> 2 boundary vertices not getting pendants.
(c) Depth-by-depth variability: no uniform bound on |π_out|
across depths.
ASSESSMENT: strict improvement over the previous cut tire definition
(no transfer overhead, cubicity restored), but the hard step
remains the same as in the partial-tire-dual framework.
Concrete next step: cut-tire analogue of tire_fiber_step2 — for
each Holton-McKay graph, build cut tire chains on both sides of a
6-cut and check R_0 ∩ R_1 empirically.
Co-Authored-By: Claude Opus 4.7 <noreply@anthropic.com>
Per user spec: instead of including the actual depth-(d±1) edges
incident to the face boundary, redefine the cut tire as:
- Face boundary walk of f (depth-d edges in H_d).
- For each vertex v on the boundary walk with degree-2 in H_d:
add a fresh vertex n_v and fresh edge {v, n_v}, labelled
"out spoke" if v has an incident depth-(d-1) edge in G'_i,
"in spoke" if v has an incident depth-(d+1) edge.
Result: each cut tire is intrinsically "cycle (or closed walk) +
labelled pendants," structurally isomorphic to the partial tire
dual D(T) from paper.tex. Pendants ↔ D(T)'s leaves, face boundary
↔ T'_ann.
This means propositions about D(T) (chromatic polynomial counts,
S_3-orbit structure, rainbow conjecture, etc.) apply verbatim to
each cut tire.
Updates:
- notes/cut_depth_label.tex: Definition rewritten, structural
remark added, table of spoke counts updated to match new defn.
- experiments/cut_tire.py: cut_tire_at() now computes labelled
pendants instead of incident edges; draw_cut_tire renders
pendant vertices (orange squares for out, green squares for in)
with edges offset toward parent-graph neighbor.
- notes/fig_cut_tire.png: regenerated.
Note grows to 6 pages.
Co-Authored-By: Claude Opus 4.7 <noreply@anthropic.com>
New note cut_tire_chain_pigeonhole.tex (3 pages) walking through the
chain pigeonhole argument applied to the cut-tire framework:
Setup: minimum counterexample G' to 4CT, 6-edge cut splits into
G'_0, G'_1, depth labelling gives chains of cut tires on each side.
Argument shape:
(1) Minimality ⇒ both G'_i have proper 3-edge-colorings.
(2) Restrict to depth-0 pendants → boundary configurations σ_0, σ_1.
(3) Glue iff σ_0 = σ_1; counterexample ⇒ no such matching.
(4) Layered description: each cut tire has inner/outer projection
constraints; adjacent tires share layers (outer of T_d =
inner of T_{d+1}).
(5) Chain pigeonhole: if each π_in is "large enough,"
R_0 ∩ R_1 ≠ ∅, contradiction.
What this needs to be a proof:
(a) Chain well-definedness: each H_d has ≥ 1 face, adjacencies
clean, no degenerate cases.
(b) Quantitative chain pigeonhole at each layer (= the rainbow
conjecture or König-lift conjecture from existing notes).
(c) Cut-tire-specific issues: H_d not cubic, face boundaries may
not be simple cycles, transfer of primal-tire results requires
verification.
Empirical chain on G'_1 of Holton-McKay #0: chain length 7,
irregular face structure across depths. No counterexample yet but
no proof either.
Net assessment: structurally sound reformulation, but inherits all
open conjectures from the existing approaches plus new technical
issues.
Concrete next step: extend tire_fiber_step2-style pairwise compatibility
check to the cut-tire setting and see if R_0 ∩ R_1 = ∅ empirically
on the Holton-McKay graphs.
Co-Authored-By: Claude Opus 4.7 <noreply@anthropic.com>
Adds Definition (cut tire) to cut_depth_label.tex:
Given the depth labelling on G'_i, for each d > 0 let H_d be the
subgraph on depth-d edges (with inherited planar embedding). For
each face f of H_d, the cut tire at (d, f) is the subgraph of G'_i
consisting of:
- every edge on the boundary walk of f (all depth d), and
- every edge of G'_i incident to the boundary walk of f with
depth d-1 or d+1.
The depth-d edges form the "face boundary"; the d-1 edges are
"inner spokes" (toward the cut); the d+1 edges are "outer spokes."
This is the dual-side analogue of the tire annular face connector
T'_{f'} (paper.tex Def. 1.16):
face boundary ↔ T'_ann (annular subgraph at depth d)
cut tire ↔ T'_{f'} (annular face connector)
inner/outer spokes ↔ inner/outer spokes of T'_{f'}
Adds experiments/cut_tire.py producing fig_cut_tire.png:
5-panel visualisation of cut tires at depths d = 1, 2, 4, 5, 6 on
G'_1 (V\S half of Holton-McKay #0). Outermost tire at d=1 (face
length 12, 5+4 spokes); innermost at d=6 (face length 12, 7+1).
Note grows from 3 → 5 pages.
Co-Authored-By: Claude Opus 4.7 <noreply@anthropic.com>
Computes a single nice layout for the full G' (Holton-McKay #0) by
trying sage-planar, sage-spring, and networkx-planar layouts and
picking the one with smallest edge-length coefficient of variation.
Spring layout wins (CV^2 = 0.049).
Then uses the SAME positions for G'_0 and G'_1, with pendant
vertices placed offset from their boundary vertex in the direction
of their cut-edge neighbor. This makes the visual correspondence
between G' and its two halves immediate.
Layout: 3 vertical panels showing G' (with cut edges highlighted),
G'_0, G'_1. Each subgraph draws only its own vertices (no orphan
vertices from the other side); all three share the same x-y limits
so positions align across panels.
Co-Authored-By: Claude Opus 4.7 <noreply@anthropic.com>
Adds a new note describing a cut-and-depth-label procedure for
the dual G' of a maximal planar G:
1. Find a 6-edge cut C in G'.
2. Remove cut edges → G'_0, G'_1.
3. In each G'_i:
a. V_i = degree-2 vertices (vertices incident to exactly 1
cut edge, hence degree 3-1=2 in induced subgraph).
b. For each v ∈ V_i, add a pendant edge to a new vertex.
Label pendants depth 0.
c. BFS-propagate: edges adjacent to a depth-d edge get
depth d+1, until all edges are labelled.
Worked example on Holton-McKay graph #0 (38-vertex non-Hamiltonian
cubic plane graph, dual of a 21-vertex triangulation):
- 128 distinct 6-edge cuts found by greedy search.
- Best matching cut: |S| = 10, cut = 6 edges with 12 distinct
endpoints (6 per side).
- G'_0: 10 + 6 = 16 vertices, max depth 2.
- G'_1: 28 + 6 = 34 vertices, max depth 7.
The procedure mirrors the 4CT cut-and-reglue reducibility scheme:
each G'_i has pendants restoring cubicity at the boundary; the
depth labels organize G'_i into concentric layers by distance to
the cut. This is the dual analogue of plane depth from a level
cycle (cf. the level-cycle generalization discussion).
Files:
experiments/cut_depth_label.py
notes/cut_depth_label.tex (3 pages)
notes/fig_cut_depth_label.png
Co-Authored-By: Claude Opus 4.7 <noreply@anthropic.com>
Extends Definition 1.2 (level edge) to also define "interlevel edge"
for the primal, and adds Definition 1.3 (level/interlevel dual edge)
classifying dual edges by whether they cross a level or interlevel
primal edge.
Useful downstream: in coloring_nested_tire_graphs, the partial tire
dual's edges can now be classified cleanly as level or interlevel
dual edges using the same vocabulary, instead of ad hoc "interior
annular edge" / "spoke edge" naming.
Paper stays at 4 pages.
Co-Authored-By: Claude Opus 4.7 <noreply@anthropic.com>
Splits the existing plane_depth_sequencing paper into two:
papers/plane_depth/paper.tex (NEW, 4 pages):
- Plane depth definition.
- Level edge, up/down/neutral triangle classification.
- Outerplanarity lemma (formerly Lemma 2.6 of PDS).
- Deep embedding G' definition.
- "Every face of G' is up or down" lemma.
- Unique level edge per face; shared level edge between adjacent faces.
- Quadrilateral decomposition definition with three types
(shallow diamond, deep diamond, S quad).
papers/plane_depth_sequencing/paper.tex (slimmed from 11 → 6 pages):
- Cites plane_depth for all foundational definitions.
- Keeps: slice, move definitions (anchor drop, level add, join,
ring completion), move selection, termination theorem.
papers/coloring_nested_tire_graphs/paper.tex:
- Bibliography updated: cite bauerfeld-depth instead of bauerfeld-pds.
- Two in-text references updated to cite the new outerplanarity
lemma in plane_depth.
Rationale: the outerplanarity / deep-embedding / quadrilateral-
decomposition material is foundational and reused by multiple
papers (and by the proposed level-cycle generalization). The
quadrilateral-sequencing programme is one specific application.
Splitting lets coloring_nested_tire_graphs cite the foundations
cleanly without dragging in the sequencing machinery.
Co-Authored-By: Claude Opus 4.7 <noreply@anthropic.com>
Attempts to prove item 1 (non-emptiness of state at L_n in closed
SR+PDS chains ending at outer triangle). Results:
PROVEN:
- S_3-closure preserved by chain propagation.
- State at L_n is either empty OR equals all 6 permutations of {1,2,3}
(the only non-empty S_3-closed subset of permutations).
- Non-emptiness propagates through intermediate tires under outward
PDS via step-1 saturation.
REMAINING GAP (conjecture, empirically true): state at L_{n-1}
intersects the "perm-paired" subset of T_n's σ_D-projection (the
σ_D values that pair with permutation σ_U). At the final step T_n
has m_n=3 < k_n, so saturation fails — chain state at L_{n-1} could
in principle lie entirely in the (non-perm-paired) parity-matching
σ_D's, but empirically doesn't.
KEY STRUCTURAL FINDING: for T=(3, k), the σ_D's paired with a
permutation σ_U equal exactly the (parity-matching σ_D's) ∩ (T's
σ_D-projection). Verified for k=3..10.
HONEST OBSERVATION: a structural proof of the remaining conjecture
(without invoking 4CT) would constitute a new proof of 4CT under
the SR+PDS modelling assumption. The chain-pigeonhole framework
reduces to this single reachability question.
Co-Authored-By: Claude Opus 4.7 <noreply@anthropic.com>
Investigation of the 'outer triangle absorption' hypothesis from
notes/outer_triangle_absorption.tex:
H2 (T_n alone absorbs anything to 6 perms): REFUTED. T_n=(3,k) alone
has σ_U-projection equal to all 27 elements of {1,2,3}^3.
H1 (chain does real work): TRUE, and structurally explained:
K_3-walk parity invariant (Lemma): in any proper edge 3-coloring
of C_n viewed as a closed walk in K_3, the 3 edge-traversal counts
all have the same parity (follows from each vertex's walk-degree
being even).
σ-color count parity (Corollary): σ at the full n cycle positions
has all-same-parity color counts.
Chain preserves parity (Theorem): forward propagation through SR
tire T=(m,k) maps state with parity matching k to state with parity
matching m, via σ_U + σ_D = σ_total with parities adding mod 2.
Outer-triangle absorption (Main Theorem): at L_n with |L_n|=3,
state has all-odd color counts summing to 3, forcing each count =
1, i.e., σ is a permutation of {1,2,3}.
Empirically verified: 0 parity violations across all chain states
in 3 representative chains (sizes 30-14643).
What's left:
- Non-emptiness: state at L_n EQUALS (not just ⊆) the 6 permutations.
Empirically yes. Likely via S_3-invariance argument.
- SR-correctness for actual G (the modeling gap, not addressed here).
If non-emptiness and SR-correctness are closed, this is a structural
proof of 4CT under the PDS framework — fundamentally different from
Birkhoff/Heesch reducibility.
Co-Authored-By: Claude Opus 4.7 <noreply@anthropic.com>
Note records the closed-chain experiment: forward-propagate state
through SR+PDS tire chains with degenerate-inner T_1 and outer-
triangle T_n (m_n=3). All 10 tested chains converge to the same
final state at L_n — exactly the 6 permutations of {1,2,3}.
Introduces the "outer triangle absorption" framing: distinguishes
H1 (chain-dependent: pigeonhole does real work to filter input to
T_n) vs H2 (T_n-only absorption: T_n's σ_U-projection is intrinsically
the 6 permutations regardless of input). Conjecture: H2 (testable by
single-tire computation).
If H2 holds, items 3-4 of the 4CT-via-tire-decomposition outline
become automatic from local data. If H1, the chain-pigeonhole does
structural work and the question is sharper.
Three-panel figure: (A) closed PDS chain (5,6,5,3) with concentric
levels and source apex; (B) outer-face dual constraint requiring
permutation-of-{1,2,3} on outer triangle; (C) state-size trajectory
showing absorption to 6 at the outer step.
Co-Authored-By: Claude Opus 4.7 <noreply@anthropic.com>
Tested 10 closed PDS chains under SR (degenerate-inner T_1, varied
middle tires, outer-triangle T_n with m_n=3). In all cases:
- Forward state grows in the middle (widest tires), shrinks toward outer.
- Final state at outer-triangle L_n has size EXACTLY 6.
- Those 6 elements are EXACTLY the permutations of {1,2,3}.
- Outer-face dual-vertex constraint (degree-3, distinct colors) is
satisfied in every chain.
This is strong empirical evidence that under SR+PDS, the entire
chain-pigeonhole story closes for 4CT:
step 1 (saturation): proven
step 2 (pairwise): automatic from step 1
step 3 (chain consistency, open): always works
step 4 (closed with outer-triangle constraint): always works,
with the 6 outer-permutations as a clean attractor.
If this holds for all internally 6-connected G under SR (likely from
Birkhoff degree ≥ 5), it's a structural proof path for 4CT for
PDS-decomposable triangulations.
Remaining: prove SR holds for all internally 6-connected G; verify
exhaustively across more chains; find symbolic proof of the
"final state = exactly 6 permutations" attractor behavior.
Co-Authored-By: Claude Opus 4.7 <noreply@anthropic.com>
Redo step 2 and step 3 under SR (no chord effect, the correct model
under PDS where O-faces are not G-faces).
Step 2 (pairwise, 14 cases): all compatible. T_1's γ-side projection
saturates {1,2,3}^γ under outward PDS (m_1 ≥ γ from step 1), so
intersection = T_2's projection, always nonempty.
Step 3 (chain consistency, 10 chains up to 6 tires): all compatible.
Forward propagation along the chain shows monotonically growing
support sizes (roughly 3x per step), never empties. Free choice
accumulates outward.
Implication: chain consistency under SR + PDS is essentially
automatic for "open" chains. The remaining gap is the boundary
condition at the outermost level (e.g., the outer triangle of a
triangulated sphere has only 6 valid σ-permutations); whether the
forward state always contains one of those is the next experiment.
Co-Authored-By: Claude Opus 4.7 <noreply@anthropic.com>
Joint-support analysis on (γ=6, T_1=(m_1=3, antipodal, SP), T_2=(k_2=3, no chord, SP)):
T_1's σ_D space = 18 elements (half of Latin set's 36; saturation-
threshold violated by m_1=3 < γ=6). T_2's σ_U space = 84 elements.
The two are intrinsically disjoint on γ, AND S_3-closure of T_1's
outer-ring colorings is already saturated (all 6 permutations
realised) — so abstract Kempe modification on the outside cannot
enlarge T_1's γ-support. The failure is A-IRREDUCIBLE in the
strict Birkhoff sense.
Significance: the SP failure cases aren't 4CT-relevant obstructions
but modeling artifacts. SP treats non-triangular O-faces as single
G-faces, which is incompatible with maximal-planar G. A faithful
maximal-planar G further triangulates these faces, changing the
face-connector and enlarging σ-supports.
Co-Authored-By: Claude Opus 4.7 <noreply@anthropic.com>
Investigated the 8 surviving triple-partitions of γ at k=k_2=9
(chord (0,3),(3,6) on B_in^(2)). Found a clean structural
description.
CLASSIFICATION of γ-edges by T_2's face structure:
For each O^(2)-face F_i, 2 γ-edges are "internal" to F_i
(their adjacent D-triangles are both in F_i).
For each adjacent face pair (F_i, F_{i+1}), 1 γ-edge is
"boundary" between them.
Total: 2r internal + r boundary = 3r γ-edges = |γ| when k=k_2.
STRUCTURAL DESCRIPTION (Prop face-pair-connection):
Latin ⊆ π_U(T_2) iff the partition has the following form:
- One block per cyclically-adjacent face pair (F_i, F_{i+1}).
- Each block = 1 boundary edge δ_{i,i+1} + 1 internal of F_i
+ 1 internal of F_{i+1}.
- For each face F_i, its 2 internal γ-edges are distributed
one per block (the two blocks involving F_i).
Count: 2^r partitions (each face has 2 choices of how to split
its internals across its 2 adjacent blocks).
AT k = k_2 = 9 (r = 3 faces): 2^3 = 8 partitions, matching the
empirical survivors.
WHY NAIVE CANDIDATES FAIL: The next-D and prev-D candidates from
worst_case_proof_sketch.tex group BOTH internals of one face into
one block (e.g., {0,1,2} = both Internal_{F_A} + δ_{AB}, no internal
F_B). This violates the "one internal per face per block" rule.
IMPLICATION: The König-lift approach can be RESCUED by replacing
the naive candidate F~_2 with any of the 2^r face-pair-connection
partitions. Apply König's theorem on bipartite face-incidence
graph of F_1 vs this new F~_2.
NEXT STEP: prove Prop face-pair-connection for all r, then apply
König lift. This is more leveraged than re-tackling the naive
construction.
Files:
experiments/k9_surviving_analysis.py
notes/k9_surviving_partitions.tex (3 pages)
Note also updates notes/induced_partition_findings.tex to point at
the new structural description.
Co-Authored-By: Claude Opus 4.7 <noreply@anthropic.com>
Tested the candidate induced γ-partition from
worst_case_proof_sketch.tex (Conj t2-induces-partition).
Findings:
1. AT k = k_2 = 6 (antipodal chord, faces 3+3): Candidate
partition (next-D or prev-D) gives Latin ⊆ π_U. ✓
But this is partly coincidental: |π_U| = 90 is so large that
ALL 10 triple-partitions of {0,..,5} have Latin ⊆ π_U.
2. AT k = k_2 = 9 (chords (0,3)(3,6), faces 3+3+3): Candidate
partition FAILS. Only 8 of all 280 triple-partitions of
{0,..,8} have Latin ⊆ π_U, and the candidate is not one of
them. The 8 surviving partitions have no obvious geometric
interpretation.
3. ASYMMETRIC k ≠ k_2 (e.g. k=6, k_2=3): Candidate doesn't
produce a triple-partition at all, and no triple-partition
has Latin ⊆ π_U. Conjecture as stated doesn't cover the
case where the empirical worst-case overlap lives.
Implication: The candidate construction is wrong past k = 6.
Step 3 (prove inclusion) is not the right next move -- we'd
be proving a false statement.
Reassessment of Approach 2: the König-overlap proposition (when
both tires give direct γ-face partitions) is still cleanly proven,
but applies to fewer cases than hoped. The asymmetric pairs that
witness the empirical worst case are not covered.
Both approaches now have known structural obstacles:
- Approach 1 (2-SAT): single open Conjecture 1.5, empirically true.
- Approach 2 (König): natural construction empirically wrong past
k=6, plus asymmetric pairs out of scope.
Honest status: chain pigeonhole has no full proof yet.
Files:
experiments/induced_partition.py
notes/induced_partition_findings.tex (3 pages)
Co-Authored-By: Claude Opus 4.7 <noreply@anthropic.com>
Adds a side-by-side comparison of the two proof attempts now in
the repo:
Approach 1 (cyclic 2-SAT, in rainbow_proof.tex):
Proves π_D = P_m (perms-per-half) for one antipodal-chord SP
tire when m_1 ≥ m - 1. Open piece: 2-SAT solvability
(Conjecture 1.5).
Approach 2 (König lift, in worst_case_proof_sketch.tex):
Proves |S_1 ∩ S_2| ≥ 6 for two adjacent SP tires sharing γ
when both chords are on γ. Open piece: T_2 induces a
γ-face partition (Conj t2-induces-partition).
Assessment: Approach 2 is more promising because (a) the hard step
is already proven (König's theorem), (b) it proves exactly what we
need (chain-pigeonhole non-emptiness, not the full π_D
characterisation), and (c) it directly explains the empirical
worst-case |S_1 ∩ S_2| = 6 = single S_3-orbit phenomenon.
Approach 1 still has value if we need finer control over π_D's
shape, but for just establishing non-empty overlap Approach 2
suffices.
Both approaches witness the SAME canonical 6-element worst-case
intersection (the rainbow S_3-orbit at γ=6 = the König-lifted
Latin S_3-orbit).
Recommended next move: attack Conj t2-induces-partition. Write
down the candidate induced γ-partition explicitly, verify it
computationally, then prove inclusion via transfer-matrix / fibre
lifting.
Note: two_approaches_comparison.tex (3 pages).
Co-Authored-By: Claude Opus 4.7 <noreply@anthropic.com>
Standalone explanation of what "2-SAT solvability" means in the
context of the rainbow proof (rainbow_proof.tex, Conjecture 1.5):
- Defines 2-SAT in general (boolean variables, 2-variable clauses,
P-time decidable).
- Maps it onto our rainbow proof: variables = orientation bits o_j
at D-positions; clauses = inter-D-position gap constraints; cyclic
chain wraps around T'_ann.
- "Solvable" ⇔ proper edge 3-coloring with given σ exists, i.e.
σ ∈ π_D.
- Cyclic 2-SAT can in principle fail; toy example (3-cycle of
not-equal clauses = odd-cycle 2-coloring obstruction).
- Empirically our system never fails for σ ∈ P_m (6-18 satisfying
orientations per σ at m=6), but a structural proof is open.
- Why it matters: proving Conjecture 1.5 upgrades the rainbow
proof's provisional corollary into a theorem and reduces chain
pigeonhole to the perms-per-half overlap.
Note: two_sat_solvability.tex (3 pages).
Co-Authored-By: Claude Opus 4.7 <noreply@anthropic.com>
Proves the clean piece: when both T1 and T2 give direct all-3 γ-face
partitions of E(γ), the worst-case overlap is ≥ 6, witnessed by
König's edge-coloring theorem on the 3-regular bipartite "face-
incidence graph" G. A proper 3-edge-coloring of G lifts to a Latin
σ on γ satisfying both face partitions, and S_3 symmetry gives 6
distinct such σ.
Identifies the gap: T2's chord is on B_in_2, not on γ, so T2 doesn't
directly give a γ-partition. The proof closes if we exhibit an
"induced γ-partition" determined by T2's annular triangulation —
conjectured but not constructed here.
Also commits chunked enumeration code and threshold-search script
launched separately.
Co-Authored-By: Claude Opus 4.7 <noreply@anthropic.com>
Final tally across γ ∈ {3,...,15,18,21}:
- 2297 total pairs tested
- 48 strict-Latin (all-3-faces) counterexamples
- 0 threshold-satisfying (m_1 ≥ γ AND k_2 ≥ γ) counterexamples (882 such pairs)
Refined conjecture (with m_1 ≥ γ precondition) holds across all tested
threshold-satisfying pairs up to γ = 12. For γ ≥ 13, threshold pairs
were unreachable under the n ≤ 27 memory cap (m_1 ≤ 27 - γ < γ).
All 48 strict-Latin CEs violate the threshold on at least one side,
consistent with the saturation behavior identified in step 1.
Co-Authored-By: Claude Opus 4.7 <noreply@anthropic.com>
didericis